On our database, most tables have a dbupddate field which indicates the datetime of the last INSERT or UPDATE applied at the row.
In order to avoid this field having an erroneous value, there exist triggers (sometimes AFTER, sometimes INSTEAD OF) which make sure that in the end, the value is correct and not whatever "manual" other value someone might try to write into that field.
Now I am performing an update statement (actualy MERGE) and I want to have an OUTPUT clause including that field. As I've read in the appropriate MS article, OUTPUT ignores triggers. 
Is there any workaround to have OUTPUT return the value dbupddate has after the triggers? I don't want to make another query to draw the info, because I am not guaranteed that in the split second between those queries, a third query of another user might not have changed quite everything.
Results after following Larnu's suggestions
I ran the examples provided, with the only exception of changing the default values of the updatetime fields to convert(datetime2,'1900-01-01') so that I might make some sense. I ran each of the 4 queries, followed by a select from their respective table and compared the updatetime values:
INSERT INTO dbo.Sample1 (Someint)
OUTPUT inserted.*
INTO @inserted
SELECT 1;
SELECT 'Sample1 INSERT',*
FROM @inserted; -- 1900-01-01 00:00:00.000000
select * from Sample1  -- 2018-11-05 13:12:13.141580
I guess the output here ignores the trigger and returns the default value that was inserted before the after trigger took effect.
DECLARE @inserted table (ID int, Someint int, updatedtime datetime2(6))
INSERT INTO dbo.Sample2 (Someint)
OUTPUT inserted.*
INTO @inserted
SELECT 1;
SELECT 'Sample2 INSERT',* --1900-01-01 00:00:00.000000
FROM @inserted;
select * from Sample2 --2018-11-05 13:12:35.580190
Same. Now the crazy part comes. I drew both the inserted and the deleted dates out:
DECLARE @updated table (ID int, Someint int, ins_updatedtime datetime2(6),del_updatedtime datetime2(6))
UPDATE dbo.Sample1 
SET Someint = 2
OUTPUT Inserted.*,Deleted.updatetime
INTO @updated;
SELECT 'Sample1 UPDATE',*
FROM @updated;   --Sample1 UPDATE   1   2   2018-11-05 13:30:01.348490  2018-11-05 13:30:01.348490
select * from Sample1  -- 1 2   2018-11-05 13:31:31.851047
DECLARE @updated table (ID int, Someint int, ins_updatedtime datetime2(6),del_updatedtime datetime2(6))
UPDATE dbo.Sample2
SET Someint = 2
OUTPUT Inserted.*,Deleted.updatetime
INTO @updated;
SELECT 'Sample2 UPDATE',* -- Sample2 UPDATE 1   2   2018-11-05 13:30:20.286422  2018-11-05 13:30:20.286422
FROM @updated;
select * from Sample2 --1   2   2018-11-05 13:31:51.679726
So, in the update cases, the default value is not present, but I have different values in the actual table and in the query's output. I know neither how to make these values the same, nor what exactly happens with the datetimes in the update case.