I'm Alan Partridge.
Best British Comedies From 2000 And Onwards. Page 2
One Foot In The Grave, AbFab, Bottom...
Waiting For God
The Detectives
Keeping Up Appearances
Birds Of A Feather
Goodnight Sweetheart
Men Behaving Badly
2point4 Children
... I'm basically just listing every even mildly-popular studio sitcom of the 1990s now. But do check them out and see if any sound of interest.
The Vicar Of Dibley
'Allo 'Allo!
My Hero
The Thin Blue Line
Red Dwarf
Also these from the 80s which are very good:
Blackadder
The Young Ones
Back to the original post-2000s request, try My Family.
Time Gentlemen Please.
The Brittas Empire (90s)
My Family
Outnumbered
Not Going Out
My Hero?
I think The IT Crowd is worth watching
Peep Show.
I must have suggested it to 50+ non-Brits during my travels, and can only remember 2 or 3 who didn't absolutely love it.