The Thick of It is probably the best one, since it's one of the best shows I've ever seen overall. Other great ones are Blackadder, Bottom and The League of Gentlemen.
I like QI and Have I Got News for You, although it's been a while since I watched either.
I started watching Ripper Street because of the hot dudes, but I haven't even finished the first season. It was alright though.
Luther never really grabbed me, and Sherlock was thoroughly meh.
And then there's a bunch of older ones of course, but idk what was or wasn't a bbc show and I can't be arsed to look it all up. Anyway, I liked AbFab, Allo Allo, Extras, The Office, Fawlty Towers, Psychoville, Inside no. 9, …