I always look forward to some of the celebrity panel shows like Mock the Week and Have I Got News for You. Although I don’t know how much I’m going to like the next season of MTW without Frankie Boyle. Sadly the sum total of my knowledge of British politics is based on these two programs.
Another vote for Spooks (Series 8 starts tomorrow night!)
Mr. Bean is very popular over here, possibly because it’s all visual humor. I once read on imdb that Rowan Atkinson himself blocked the shows from appearing in Italy, because he liked to takes his holidays there and wanted to remain unrecognized. That entry is not there anymore, so I don’t know if he changed his mind or it was false to begin with.
We’ve seen three of the Black Adders, the one during King George’s time the missing one. In the fourth series, WWI, the wife really liked that there was a character named Darling. You can imagine her joy at discovering the present British chancellor of the Exchequer is Alistair Darling. 
Blackadder III has one of my favorite lines ever (spoiler):
Prince George: What can I do to a woman that I can’t do to you?
Blackadder: I cannot conceive, sir.
No love for the Young Ones? Hands down, one of the funniest shows I have ever watched. 4 university students share a house, and various wacky and surrealistic hijinks ensue.
It predates Absolutely Fabulous and is almost nothing like it (besides being hilarious), but the two shows shared several actors.
He was also brilliant in in Hardware, a series with Martin Freeman. Vastly underrated and in my opinion pretty comical. Though I was always a sucker for Ken Morely being silly. And he was never in anything where he wasn’t silly. I think they only made one or two series of it.
Set in a hardware shop that was always under threat from the local megahugegigantico DIY superstore up the road.
For old school classic comedy, you can’t go far wrong with Porridge and Open All Hours. Both starring the master himself Mr Ronnie Barker with assistance from a youthful David Jason. Though in Porridge his youth was fairly well camouflaged.
For those of us interested in Hollywood glamour and blockbusters, in Porridge, the callow youth Godber is played by Richard Beckinsale who was the father of Kate Beckinsale. He died very young, but as well as being brilliant in Porridge, he was also in another comical old school series - Rising Damp.
Seeing as Steve Coogan has been mentioned a few times, what about Sunshine? Comedy and tragedy together. And Coogan’s mates in that were none other than Craig Cash and Phil Mealy, who gave us Early Doors. If you like Sunshine, you’ll like Early Doors and The Royle Family. If you don’t, you won’t.
I’d totally forgotten David Jason was in Porridge!
Only 31 when he died - a great loss to UK acting.
My favourite comedy series is Red Dwarf. It might be a little long in the tooth now but if I could wipe my mind it would be first on my list to re-watch.
Other honourable mentions:
Black Books. Bill Bailey and Dylan Moran are fantastic in their own right and play off each other brilliantly. The third series is a bit shit but the first two are great.
I enjoy Blackadder and lots of people love it.
The I.T. Crowd is fantastic, as is The Mighty Boosh.
Other comedy programs that I will watch if I find them on but wont seek out: Father Ted, Young Ones and Fawlty Towers.
I’m not sure if you have any access to British radio shows but if you do then I recommend Old Harry’s Game, which features Andy Hamilton as the devil trying to get mankind to behave itself due to hell overcrowding. That Mitchell and Webb Sound is a sketch show that’s much better than the TV series. Revolting People is a comedy set in Baltimore just before the American civil war, It also features Andy Hamilton, which is how I was introduced to it.
I remember The Young Ones very well. It played in the US back in the early days of MTV.
And Young Ones actors Rik Mayall and Adrian Edmondson both starred in the excellent Bottom, which we also watched courtesy of our English friend. The wife hated Bottom but could not tear her eyes away, sort of like you do at a bad traffic accident. I thought it was great.
And then there was their Filthy Rich & Catflap, which was almost as good and also featured another Young Ones veteran, Nigel Planer. Saw that one, too.
How can that be remotely possible, with your name?
You, Baron Greenback, are the arch enemy and sworn nemesis of David Jason’s very finest creation in a long career of iconic characterisiations that are woven into very fabric of modern British life.
God, I love Danger Mouse.
So, if I was to casually mention Super Ted and Bananaman in casual conversation, you’d get some of those Happy Childhood Memories™? 
I always felt that the crap sitcom in Extras was a dig at Hardware and other comedies like that.
Most of the ones I’d recommend have been covered already.
I’m a bit miffed at all the Peep Show love. I never warmed to it at all. The Office, as cringeworthy as it was, always made me laugh. But Peep Show, for me just has the cringe and none of the laughs.
More recent series I enjoyed have been The Thick of It and Being Human.
I don’t think anyone’s mentioned The Smoking Room yet. I missed it when it first came out but I’ve been catching up on it recently. Its set entirely within the dingy smoking room in an office building, with various characters drifting in and out to have a ciggy and/or avoid doing any actual work. It stars Peep Show’s Robert Webb. It’s all talk, no action, you hear of but never see events outside the room; it sometimes feels like a low-IQ comedy version of 12 Angry Men.
And, just to make this whole thing a little bit geekier, Planer is heavily tied into Discworld. He does the reading for the audiobooks and appeared in both Hogfather and The Colour of Magic.
*Excellent. The brainwashed colonel has send his greatest agent to his doom. And when that dim-witted desperately deluded duo reach the end of their wild goose chase, they will discover, the wildest, wickedest goose the world has ever seen!!!
*
To be fair when I changed my username I asked Tuba about all sorts of variations on Danger Mouse, but they were all taken.
So then, I turned to the dark side…
Croaks
If you are looking for British comedies, then you have to check out Fawlty Towers. It is widely accepted as one of the best, if not the best, sitcom. And there is a BBC Video Fawlty Towers Remastered ‘lookalike contest’ where you can win a SIGNED DVD from John Cleese, with winners being selected by the man himself! If you’re game, head over to http://tr.im/fawltycontest before 11/15 to get in on this bloody awesome opportunity!
Is Doc Martin classified as a comedy? Perhaps it’s more of a dramedy.
Whatever it is, I love it. Martin Clunes is as funny as hell in his role as a stodgy, socially clueless small village doctor. I only recently learned that he’s Jeremy Brett’s cousin, and got into acting as a result of Brett’s encouragement.
I recall PBS running Fawlty Towers in the US way back when. From time to time, I stumble upon it showing in a bar or pub here, usually one that’s British-owned.
What a great thread. W/regard to the OP, I also loved the premise of Life on Mars but don’t feel it stands the test of time and Ashes to Ashes was instant sharkbait.
I’ll second Dr. Martin and proffer Men behaving badly.
On the drama front I would recommend to you City of Vice. As it’s a channel 4 show I imagine it can be found using the 4OD link upthread.
It’s basically the very early days of introducing lawmen to London in the 1700’s. Filthy, depraved and enormous fun
Tell me it gets better than the first episode and I’ll try it again. I couldn’t make it through the first one. I was trying to be completist and starting from the beginning.
The second, third, and fourth series are, IMHO, the “best”, but otherwise I think they’re all pretty good. Series II is pretty much set solely aboard the Red Dwarf, but they spend a mixture of time on and off the ship in series III-IV, spend most of their time in Starbug in series V (followed by the entirety of their time being aboard Starbug in series VI and VII…) series VIII is back on Red Dwarf* and off it as well, there is no series IX (don’t ask), and series X consists so far of three episodes, although they have apparently greenlit the rest of the series and it should start shooting next year.
Or, to put it another way: Give it another go; you could even try starting at series II or series III (which are bit more “interesting” than the first series, which is really about the interaction between Lister and Rimmer) and see how you go from there.
*Yes, I know. Don’t complicate it for the non-Dwarfers, OK?