Even though I don't get many British comedies here in America, from the ones that I have watched on BBC America I can say that I absolutely love: The Vicar of Dibley, Top Gear (I consider it a comedy ), Outnumbered, The Inbetweeners, The Graham Norton Show, and Come Fly With Me.
The official genre of it is a "Science fiction, Comedy-drama" It makes me laugh all the time, that's what a comedy is. o3o
Any English Comedy ,because that was Britain does best, Action Movies and shows is What America does best. When America a British comedy they wreck it eg Red Dwarf British = Funny, American = rubbish. But you will never get good Action shows like 24, the cape, CSI, Human Target e.t.c in Britain because We don't to action, we do funny and Drama, however some American Comedy is Epic, Family Guy, South Park, e.t.c.
My top 5 favorites have always been: Are You Being Served? Keeping Up Appearances Coupling As Time Goes By May to December