Current M.2 drive with a Windows 10 install is 128GB. New M.2 drive is 1TB.
I only have one M.2 slot.
Question: If I clone the old drive to an external hard drive that is 2TB, will it cause any issues when I clone to the new 1TB M.2 due to size difference? Or should I make a 128GB partition on external hard drive to clone/restore file system from?
If you wish to avoid problems, back up the data on the small drive, remove it entirely. Install the new SSD, install Windows and restore data. Your overall partitioning on the main disk will be better this way.
Trying to use both drives at once in some way may require you to have it that way going forward.
Just avoid all the problems and do it one step at a time as I laid out in the original paragraph here.
Once done you can use external drives as you wish without any issue.
I also avoid cloning when making a new installation because cloning brings over old problems.
I have done things as I have suggested for many years with excellent long term results.