A lot of the comedies that I think have aged really well have already been mentioned:
Yes Minister, Yes Prime Minister
Outnumbered
Fawlty Towers
Mitchell and Webb
Black Adder
The Royle Family
The Office
I'm Alan Partridge
Father Ted
The Thick of It
Porridge
Spaced
I don't think anyone's mentioned Extras or Marion and Geoff but I think they're still funny and very watchable.
I got sidetracked by the reddit thread that was quoted earlier and googled the Mitchell and Webb Sherlock Holmes sketch which was mentioned. I don't remember it but think it's brilliant.
I'm not sure about Frasier. It's very witty but he's also pretty sleazy and there was some slut shaming of Ros which I never liked.
Allo, Allo has been mentioned but I thought that was bloody awful and very unfunny at the time. I did like Secret Army, the series that it was a pastiche of.
A drama that I think holds up really well is Tenko. I loved watching it with my mum when it first came out and then, a couple of decades later, I rewatched it with my own dd and she loved it too.
I'm currently watching Clocking Off. It's 25 years old and the phones and computers are dated but nothing else is. I didn't see it first time round and am absolutely loving it.
It has a similar feel to The Street (ensemble cast, ongoing storylines but focusing on different characters each episode) which was made a few years later and which I enjoyed at the time.
Once I've finished bingeing Clocking Off, I'm going to see if The Street is available anywhere and hope that's also held up well.
Alan Bennett's Talking Heads monologues are wonderful. The original ones from the 80s were so well done that I didn't bother with the 2020 remake. That seemed utterly pointless.