I binged Gavin and Stacy last Spring - James Corden is one of the creators and writers. I think it is still on Amazon.
I have Every Stoopid Episode
You know, Mrs. L.A. did binge-watch the entire series of Monty Python’s Flying Circus with me.
Boxed set. 
Also, Space:1999.
I believe that the “Bambi” episode of the Young Ones is second to no other television episode ever in the history of television. And the rest of the episodes are wonderful, too. And Bottom.
This is unduly abrasive for Cafe Society. If your don’t like Father Brown, just say so. Don’t make it about the poster who does like it.
I loved Dave Allen at Large. Not long before he died in 2005, he said “I still think of myself as I was 25 years ago. Then I look in a mirror and see an old bastard, and I realise it’s me.”
In the late-'70s, Dave Allen At Large was on PBS, followed by Benny Hill. I much preferred Dave Allen At Large. Though I haven’t seen the show in 40 years, I still laugh at his skits when they come to mind.
If you think the Inspector Morse character was horrible in the TV series, check out the original books, he was worse. Treated Lewis like shit.
FWIW I enjoyed Father Brown. Some of the early episodes are based on the original Chesterton stories.
As it goes on into later seasons some of the recurring tropes get a little tired (whatshername finds the dead body and screams, the police inspector is grumpy and dismissive, etc) but Mark Williams does a great job IMHO.
I can normally go several months without watching ITV. But they have produced one good comedy in the last few years. Plebs
A few words of caution:
Peep Show is possibly my favorite show ever, but I didn’t realize how tied it is to British culture and idioms. I had to explain almost every joke to my GF, and we gave up after one episode.
Jonathan Creek : Season 5, the last, was dreadful. Not Season 4 of Sherlock bad, but still, don’t bother with it. I’m saying this because I know some online services recommend the most recent season first.
Earlier seasons are great though.
The IT Crowd : Gets recommended a lot, but rewatching some episodes lately, I felt it hadn’t aged well. It doesn’t compare well with some of the tech shows that have come since, like Silicon Valley.
And just so I am not only on a negative tip:
The British version of House of Cards, from way back in the 1990s, is still a great watch. It’s just a shame it’s only 6 episodes (IIRC).
I’ve not seen it streamed (but you never know your luck) but find Dalziel and Pascoe, which is a British police procedural, but closely focused on the relationship between the two characters. I know this makes it sound like Cagney and Lacey, but its much better than that. And starred Warren Clarke. Based on an excellent series of books by Reginald Hill, they drifted off canon at some point and lost me a bit.
The TV adaptation of the Rebus novels didn’t do much for me. It starred John Hannah as Rebus, but he was also in the rather fantastic cop parody A Touch of Cloth.
Next of Kin with Penelope Keith. Grandparents have to assume the raising of their grandchildren. Battle of the generations doesn’t get much better than this.
I do like the differences in Antiques Roadshow. It’s great when someone from the Great Britain is told that their “whatever” is about 500 years old and VERY valuable. The response is usually a quiet “Oh, that’s lovely.”
Nor me (John Hannah was too young and glossy for the part). There were more later, starring Ken Stott, who was a much better fit, in look and manner, for the character as in the novels.
He was also on the other end of an investigation in The Missing, also worth watching, as was its spin-off Baptiste.
The follow-on shows (To Play The King and The Final Cut) were worthwhile as well.
If you want something uniquely british, then try some of Alan Bleasdales work. GBH covers a period of the 80s under Thatcher, where the government is trying to bring down a left wing council in England. I suspect some of the dirty tricks covered are still valid.
Another in the same vein is A very british coup, a fictional idea that a left wing government is elected in the UK (and that was possible in the 80s, Thatcher did struggle at times being popular), and it aligning more with the USSR than an over zealous Reagan at the time.
There was also The Amazing Mrs Pritchard, in which a supermarket manager stood for election and ended up as Prime Minister.
Definitely worth a look - like a British ‘Police Squad!’ in its humour, written by Charlie Brooker.
I concur. Yes, the later episodes are over formulaic, but that happens with a lot of shows.
Dalziel and Pascoe is on Britbox.
And “very valuable” is usually considerably lower in value than American “very valuable”.