I wrote a full replacement for sys.stderr and just duplicated the code renaming stderr to stdout to make it also available to replace sys.stdout.
To do this I create the same object type as the current stderr and stdout, and forward all methods to the original system stderr and stdout:
import os
import sys
import logging
class StdErrReplament(object):
    """
        How to redirect stdout and stderr to logger in Python
        https://stackoverflow.com/questions/19425736/how-to-redirect-stdout-and-stderr-to-logger-in-python
        Set a Read-Only Attribute in Python?
        https://stackoverflow.com/questions/24497316/set-a-read-only-attribute-in-python
    """
    is_active = False
    @classmethod
    def lock(cls, logger):
        """
            Attach this singleton logger to the `sys.stderr` permanently.
        """
        global _stderr_singleton
        global _stderr_default
        global _stderr_default_class_type
        # On Sublime Text, `sys.__stderr__` is set to None, because they already replaced `sys.stderr`
        # by some `_LogWriter()` class, then just save the current one over there.
        if not sys.__stderr__:
            sys.__stderr__ = sys.stderr
        try:
            _stderr_default
            _stderr_default_class_type
        except NameError:
            _stderr_default = sys.stderr
            _stderr_default_class_type = type( _stderr_default )
        # Recreate the sys.stderr logger when it was reset by `unlock()`
        if not cls.is_active:
            cls.is_active = True
            _stderr_write = _stderr_default.write
            logger_call = logger.debug
            clean_formatter = logger.clean_formatter
            global _sys_stderr_write
            global _sys_stderr_write_hidden
            if sys.version_info <= (3,2):
                logger.file_handler.terminator = '\n'
            # Always recreate/override the internal write function used by `_sys_stderr_write`
            def _sys_stderr_write_hidden(*args, **kwargs):
                """
                    Suppress newline in Python logging module
                    https://stackoverflow.com/questions/7168790/suppress-newline-in-python-logging-module
                """
                try:
                    _stderr_write( *args, **kwargs )
                    file_handler = logger.file_handler
                    formatter = file_handler.formatter
                    terminator = file_handler.terminator
                    file_handler.formatter = clean_formatter
                    file_handler.terminator = ""
                    kwargs['extra'] = {'_duplicated_from_file': True}
                    logger_call( *args, **kwargs )
                    file_handler.formatter = formatter
                    file_handler.terminator = terminator
                except Exception:
                    logger.exception( "Could not write to the file_handler: %s(%s)", file_handler, logger )
                    cls.unlock()
            # Only create one `_sys_stderr_write` function pointer ever
            try:
                _sys_stderr_write
            except NameError:
                def _sys_stderr_write(*args, **kwargs):
                    """
                        Hides the actual function pointer. This allow the external function pointer to
                        be cached while the internal written can be exchanged between the standard
                        `sys.stderr.write` and our custom wrapper around it.
                    """
                    _sys_stderr_write_hidden( *args, **kwargs )
        try:
            # Only create one singleton instance ever
            _stderr_singleton
        except NameError:
            class StdErrReplamentHidden(_stderr_default_class_type):
                """
                    Which special methods bypasses __getattribute__ in Python?
                    https://stackoverflow.com/questions/12872695/which-special-methods-bypasses-getattribute-in-python
                """
                if hasattr( _stderr_default, "__abstractmethods__" ):
                    __abstractmethods__ = _stderr_default.__abstractmethods__
                if hasattr( _stderr_default, "__base__" ):
                    __base__ = _stderr_default.__base__
                if hasattr( _stderr_default, "__bases__" ):
                    __bases__ = _stderr_default.__bases__
                if hasattr( _stderr_default, "__basicsize__" ):
                    __basicsize__ = _stderr_default.__basicsize__
                if hasattr( _stderr_default, "__call__" ):
                    __call__ = _stderr_default.__call__
                if hasattr( _stderr_default, "__class__" ):
                    __class__ = _stderr_default.__class__
                if hasattr( _stderr_default, "__delattr__" ):
                    __delattr__ = _stderr_default.__delattr__
                if hasattr( _stderr_default, "__dict__" ):
                    __dict__ = _stderr_default.__dict__
                if hasattr( _stderr_default, "__dictoffset__" ):
                    __dictoffset__ = _stderr_default.__dictoffset__
                if hasattr( _stderr_default, "__dir__" ):
                    __dir__ = _stderr_default.__dir__
                if hasattr( _stderr_default, "__doc__" ):
                    __doc__ = _stderr_default.__doc__
                if hasattr( _stderr_default, "__eq__" ):
                    __eq__ = _stderr_default.__eq__
                if hasattr( _stderr_default, "__flags__" ):
                    __flags__ = _stderr_default.__flags__
                if hasattr( _stderr_default, "__format__" ):
                    __format__ = _stderr_default.__format__
                if hasattr( _stderr_default, "__ge__" ):
                    __ge__ = _stderr_default.__ge__
                if hasattr( _stderr_default, "__getattribute__" ):
                    __getattribute__ = _stderr_default.__getattribute__
                if hasattr( _stderr_default, "__gt__" ):
                    __gt__ = _stderr_default.__gt__
                if hasattr( _stderr_default, "__hash__" ):
                    __hash__ = _stderr_default.__hash__
                if hasattr( _stderr_default, "__init__" ):
                    __init__ = _stderr_default.__init__
                if hasattr( _stderr_default, "__init_subclass__" ):
                    __init_subclass__ = _stderr_default.__init_subclass__
                if hasattr( _stderr_default, "__instancecheck__" ):
                    __instancecheck__ = _stderr_default.__instancecheck__
                if hasattr( _stderr_default, "__itemsize__" ):
                    __itemsize__ = _stderr_default.__itemsize__
                if hasattr( _stderr_default, "__le__" ):
                    __le__ = _stderr_default.__le__
                if hasattr( _stderr_default, "__lt__" ):
                    __lt__ = _stderr_default.__lt__
                if hasattr( _stderr_default, "__module__" ):
                    __module__ = _stderr_default.__module__
                if hasattr( _stderr_default, "__mro__" ):
                    __mro__ = _stderr_default.__mro__
                if hasattr( _stderr_default, "__name__" ):
                    __name__ = _stderr_default.__name__
                if hasattr( _stderr_default, "__ne__" ):
                    __ne__ = _stderr_default.__ne__
                if hasattr( _stderr_default, "__new__" ):
                    __new__ = _stderr_default.__new__
                if hasattr( _stderr_default, "__prepare__" ):
                    __prepare__ = _stderr_default.__prepare__
                if hasattr( _stderr_default, "__qualname__" ):
                    __qualname__ = _stderr_default.__qualname__
                if hasattr( _stderr_default, "__reduce__" ):
                    __reduce__ = _stderr_default.__reduce__
                if hasattr( _stderr_default, "__reduce_ex__" ):
                    __reduce_ex__ = _stderr_default.__reduce_ex__
                if hasattr( _stderr_default, "__repr__" ):
                    __repr__ = _stderr_default.__repr__
                if hasattr( _stderr_default, "__setattr__" ):
                    __setattr__ = _stderr_default.__setattr__
                if hasattr( _stderr_default, "__sizeof__" ):
                    __sizeof__ = _stderr_default.__sizeof__
                if hasattr( _stderr_default, "__str__" ):
                    __str__ = _stderr_default.__str__
                if hasattr( _stderr_default, "__subclasscheck__" ):
                    __subclasscheck__ = _stderr_default.__subclasscheck__
                if hasattr( _stderr_default, "__subclasses__" ):
                    __subclasses__ = _stderr_default.__subclasses__
                if hasattr( _stderr_default, "__subclasshook__" ):
                    __subclasshook__ = _stderr_default.__subclasshook__
                if hasattr( _stderr_default, "__text_signature__" ):
                    __text_signature__ = _stderr_default.__text_signature__
                if hasattr( _stderr_default, "__weakrefoffset__" ):
                    __weakrefoffset__ = _stderr_default.__weakrefoffset__
                if hasattr( _stderr_default, "mro" ):
                    mro = _stderr_default.mro
                def __init__(self):
                    """
                        Override any super class `type( _stderr_default )` constructor, so we can 
                        instantiate any kind of `sys.stderr` replacement object, in case it was already 
                        replaced by something else like on Sublime Text with `_LogWriter()`.
                        Assures all attributes were statically replaced just above. This should happen in case
                        some new attribute is added to the python language.
                        This also ignores the only two methods which are not equal, `__init__()` and `__getattribute__()`.
                    """
                    different_methods = ("__init__", "__getattribute__")
                    attributes_to_check = set( dir( object ) + dir( type ) )
                    for attribute in attributes_to_check:
                        if attribute not in different_methods \
                                and hasattr( _stderr_default, attribute ):
                            base_class_attribute = super( _stderr_default_class_type, self ).__getattribute__( attribute )
                            target_class_attribute = _stderr_default.__getattribute__( attribute )
                            if base_class_attribute != target_class_attribute:
                                sys.stderr.write( "    The base class attribute `%s` is different from the target class:\n%s\n%s\n\n" % (
                                        attribute, base_class_attribute, target_class_attribute ) )
                def __getattribute__(self, item):
                    if item == 'write':
                        return _sys_stderr_write
                    try:
                        return _stderr_default.__getattribute__( item )
                    except AttributeError:
                        return super( _stderr_default_class_type, _stderr_default ).__getattribute__( item )
            _stderr_singleton = StdErrReplamentHidden()
            sys.stderr = _stderr_singleton
        return cls
    @classmethod
    def unlock(cls):
        """
            Detach this `stderr` writer from `sys.stderr` and allow the next call to `lock()` create
            a new writer for the stderr.
        """
        if cls.is_active:
            global _sys_stderr_write_hidden
            cls.is_active = False
            _sys_stderr_write_hidden = _stderr_default.write
class StdOutReplament(object):
    """
        How to redirect stdout and stderr to logger in Python
        https://stackoverflow.com/questions/19425736/how-to-redirect-stdout-and-stderr-to-logger-in-python
        Set a Read-Only Attribute in Python?
        https://stackoverflow.com/questions/24497316/set-a-read-only-attribute-in-python
    """
    is_active = False
    @classmethod
    def lock(cls, logger):
        """
            Attach this singleton logger to the `sys.stdout` permanently.
        """
        global _stdout_singleton
        global _stdout_default
        global _stdout_default_class_type
        # On Sublime Text, `sys.__stdout__` is set to None, because they already replaced `sys.stdout`
        # by some `_LogWriter()` class, then just save the current one over there.
        if not sys.__stdout__:
            sys.__stdout__ = sys.stdout
        try:
            _stdout_default
            _stdout_default_class_type
        except NameError:
            _stdout_default = sys.stdout
            _stdout_default_class_type = type( _stdout_default )
        # Recreate the sys.stdout logger when it was reset by `unlock()`
        if not cls.is_active:
            cls.is_active = True
            _stdout_write = _stdout_default.write
            logger_call = logger.debug
            clean_formatter = logger.clean_formatter
            global _sys_stdout_write
            global _sys_stdout_write_hidden
            if sys.version_info <= (3,2):
                logger.file_handler.terminator = '\n'
            # Always recreate/override the internal write function used by `_sys_stdout_write`
            def _sys_stdout_write_hidden(*args, **kwargs):
                """
                    Suppress newline in Python logging module
                    https://stackoverflow.com/questions/7168790/suppress-newline-in-python-logging-module
                """
                try:
                    _stdout_write( *args, **kwargs )
                    file_handler = logger.file_handler
                    formatter = file_handler.formatter
                    terminator = file_handler.terminator
                    file_handler.formatter = clean_formatter
                    file_handler.terminator = ""
                    kwargs['extra'] = {'_duplicated_from_file': True}
                    logger_call( *args, **kwargs )
                    file_handler.formatter = formatter
                    file_handler.terminator = terminator
                except Exception:
                    logger.exception( "Could not write to the file_handler: %s(%s)", file_handler, logger )
                    cls.unlock()
            # Only create one `_sys_stdout_write` function pointer ever
            try:
                _sys_stdout_write
            except NameError:
                def _sys_stdout_write(*args, **kwargs):
                    """
                        Hides the actual function pointer. This allow the external function pointer to
                        be cached while the internal written can be exchanged between the standard
                        `sys.stdout.write` and our custom wrapper around it.
                    """
                    _sys_stdout_write_hidden( *args, **kwargs )
        try:
            # Only create one singleton instance ever
            _stdout_singleton
        except NameError:
            class StdOutReplamentHidden(_stdout_default_class_type):
                """
                    Which special methods bypasses __getattribute__ in Python?
                    https://stackoverflow.com/questions/12872695/which-special-methods-bypasses-getattribute-in-python
                """
                if hasattr( _stdout_default, "__abstractmethods__" ):
                    __abstractmethods__ = _stdout_default.__abstractmethods__
                if hasattr( _stdout_default, "__base__" ):
                    __base__ = _stdout_default.__base__
                if hasattr( _stdout_default, "__bases__" ):
                    __bases__ = _stdout_default.__bases__
                if hasattr( _stdout_default, "__basicsize__" ):
                    __basicsize__ = _stdout_default.__basicsize__
                if hasattr( _stdout_default, "__call__" ):
                    __call__ = _stdout_default.__call__
                if hasattr( _stdout_default, "__class__" ):
                    __class__ = _stdout_default.__class__
                if hasattr( _stdout_default, "__delattr__" ):
                    __delattr__ = _stdout_default.__delattr__
                if hasattr( _stdout_default, "__dict__" ):
                    __dict__ = _stdout_default.__dict__
                if hasattr( _stdout_default, "__dictoffset__" ):
                    __dictoffset__ = _stdout_default.__dictoffset__
                if hasattr( _stdout_default, "__dir__" ):
                    __dir__ = _stdout_default.__dir__
                if hasattr( _stdout_default, "__doc__" ):
                    __doc__ = _stdout_default.__doc__
                if hasattr( _stdout_default, "__eq__" ):
                    __eq__ = _stdout_default.__eq__
                if hasattr( _stdout_default, "__flags__" ):
                    __flags__ = _stdout_default.__flags__
                if hasattr( _stdout_default, "__format__" ):
                    __format__ = _stdout_default.__format__
                if hasattr( _stdout_default, "__ge__" ):
                    __ge__ = _stdout_default.__ge__
                if hasattr( _stdout_default, "__getattribute__" ):
                    __getattribute__ = _stdout_default.__getattribute__
                if hasattr( _stdout_default, "__gt__" ):
                    __gt__ = _stdout_default.__gt__
                if hasattr( _stdout_default, "__hash__" ):
                    __hash__ = _stdout_default.__hash__
                if hasattr( _stdout_default, "__init__" ):
                    __init__ = _stdout_default.__init__
                if hasattr( _stdout_default, "__init_subclass__" ):
                    __init_subclass__ = _stdout_default.__init_subclass__
                if hasattr( _stdout_default, "__instancecheck__" ):
                    __instancecheck__ = _stdout_default.__instancecheck__
                if hasattr( _stdout_default, "__itemsize__" ):
                    __itemsize__ = _stdout_default.__itemsize__
                if hasattr( _stdout_default, "__le__" ):
                    __le__ = _stdout_default.__le__
                if hasattr( _stdout_default, "__lt__" ):
                    __lt__ = _stdout_default.__lt__
                if hasattr( _stdout_default, "__module__" ):
                    __module__ = _stdout_default.__module__
                if hasattr( _stdout_default, "__mro__" ):
                    __mro__ = _stdout_default.__mro__
                if hasattr( _stdout_default, "__name__" ):
                    __name__ = _stdout_default.__name__
                if hasattr( _stdout_default, "__ne__" ):
                    __ne__ = _stdout_default.__ne__
                if hasattr( _stdout_default, "__new__" ):
                    __new__ = _stdout_default.__new__
                if hasattr( _stdout_default, "__prepare__" ):
                    __prepare__ = _stdout_default.__prepare__
                if hasattr( _stdout_default, "__qualname__" ):
                    __qualname__ = _stdout_default.__qualname__
                if hasattr( _stdout_default, "__reduce__" ):
                    __reduce__ = _stdout_default.__reduce__
                if hasattr( _stdout_default, "__reduce_ex__" ):
                    __reduce_ex__ = _stdout_default.__reduce_ex__
                if hasattr( _stdout_default, "__repr__" ):
                    __repr__ = _stdout_default.__repr__
                if hasattr( _stdout_default, "__setattr__" ):
                    __setattr__ = _stdout_default.__setattr__
                if hasattr( _stdout_default, "__sizeof__" ):
                    __sizeof__ = _stdout_default.__sizeof__
                if hasattr( _stdout_default, "__str__" ):
                    __str__ = _stdout_default.__str__
                if hasattr( _stdout_default, "__subclasscheck__" ):
                    __subclasscheck__ = _stdout_default.__subclasscheck__
                if hasattr( _stdout_default, "__subclasses__" ):
                    __subclasses__ = _stdout_default.__subclasses__
                if hasattr( _stdout_default, "__subclasshook__" ):
                    __subclasshook__ = _stdout_default.__subclasshook__
                if hasattr( _stdout_default, "__text_signature__" ):
                    __text_signature__ = _stdout_default.__text_signature__
                if hasattr( _stdout_default, "__weakrefoffset__" ):
                    __weakrefoffset__ = _stdout_default.__weakrefoffset__
                if hasattr( _stdout_default, "mro" ):
                    mro = _stdout_default.mro
                def __init__(self):
                    """
                        Override any super class `type( _stdout_default )` constructor, so we can 
                        instantiate any kind of `sys.stdout` replacement object, in case it was already 
                        replaced by something else like on Sublime Text with `_LogWriter()`.
                        Assures all attributes were statically replaced just above. This should happen in case
                        some new attribute is added to the python language.
                        This also ignores the only two methods which are not equal, `__init__()` and `__getattribute__()`.
                    """
                    different_methods = ("__init__", "__getattribute__")
                    attributes_to_check = set( dir( object ) + dir( type ) )
                    for attribute in attributes_to_check:
                        if attribute not in different_methods \
                                and hasattr( _stdout_default, attribute ):
                            base_class_attribute = super( _stdout_default_class_type, self ).__getattribute__( attribute )
                            target_class_attribute = _stdout_default.__getattribute__( attribute )
                            if base_class_attribute != target_class_attribute:
                                sys.stdout.write( "    The base class attribute `%s` is different from the target class:\n%s\n%s\n\n" % (
                                        attribute, base_class_attribute, target_class_attribute ) )
                def __getattribute__(self, item):
                    if item == 'write':
                        return _sys_stdout_write
                    try:
                        return _stdout_default.__getattribute__( item )
                    except AttributeError:
                        return super( _stdout_default_class_type, _stdout_default ).__getattribute__( item )
            _stdout_singleton = StdOutReplamentHidden()
            sys.stdout = _stdout_singleton
        return cls
    @classmethod
    def unlock(cls):
        """
            Detach this `stdout` writer from `sys.stdout` and allow the next call to `lock()` create
            a new writer for the stdout.
        """
        if cls.is_active:
            global _sys_stdout_write_hidden
            cls.is_active = False
            _sys_stdout_write_hidden = _stdout_default.write
To use this you can just call StdErrReplament::lock(logger) and StdOutReplament::lock(logger)
passing the logger you want to use to send the output text. For example:
import os
import sys
import logging
current_folder = os.path.dirname( os.path.realpath( __file__ ) )
log_file_path = os.path.join( current_folder, "my_log_file.txt" )
file_handler = logging.FileHandler( log_file_path, 'a' )
file_handler.formatter = logging.Formatter( "%(asctime)s %(name)s %(levelname)s - %(message)s", "%Y-%m-%d %H:%M:%S" )
log = logging.getLogger( __name__ )
log.setLevel( "DEBUG" )
log.addHandler( file_handler )
log.file_handler = file_handler
log.clean_formatter = logging.Formatter( "", "" )
StdOutReplament.lock( log )
StdErrReplament.lock( log )
log.debug( "I am doing usual logging debug..." )
sys.stderr.write( "Tests 1...\n" )
sys.stdout.write( "Tests 2...\n" )
Running this code, you will see on the screen: 

And on the file contents:

If you would like to also see the contents of the log.debug calls on the screen, you will need to add a stream handler to your logger. On this case it would be like this:
import os
import sys
import logging
class ContextFilter(logging.Filter):
    """ This filter avoids duplicated information to be displayed to the StreamHandler log. """
    def filter(self, record):
        return not "_duplicated_from_file" in record.__dict__
current_folder = os.path.dirname( os.path.realpath( __file__ ) )
log_file_path = os.path.join( current_folder, "my_log_file.txt" )
stream_handler = logging.StreamHandler()
file_handler = logging.FileHandler( log_file_path, 'a' )
formatter = logging.Formatter( "%(asctime)s %(name)s %(levelname)s - %(message)s", "%Y-%m-%d %H:%M:%S" )
file_handler.formatter = formatter
stream_handler.formatter = formatter
stream_handler.addFilter( ContextFilter() )
log = logging.getLogger( __name__ )
log.setLevel( "DEBUG" )
log.addHandler( file_handler )
log.addHandler( stream_handler )
log.file_handler = file_handler
log.stream_handler = stream_handler
log.clean_formatter = logging.Formatter( "", "" )
StdOutReplament.lock( log )
StdErrReplament.lock( log )
log.debug( "I am doing usual logging debug..." )
sys.stderr.write( "Tests 1...\n" )
sys.stdout.write( "Tests 2...\n" )
Which would output like this when running:

While it would still saving this to the file my_log_file.txt:

When disabling this with StdErrReplament:unlock(), it will only restore the standard behavior of the stderr stream, as the attached logger cannot be never detached because someone else can have a reference to its older version. This is why it is a global singleton which can never dies. Therefore, in case of reloading this module with imp or something else, it will never recapture the current sys.stderr as it was already injected on it and have it saved internally.