It was the comedies that were particularly good and very British. Some were very unusual and bizarre, the late night shows. But they were also where writers and comedians got a break and then became mainstream. I would guess that kind of thing is now made for the internet, and its a shame to see everything go so niche.
An article I came across a couple of years ago (wish I could find it!) talked about how there was this period of time when British TV started to diversify the source of talent, around the 80s and 90s. You got shows like Red Dwarf where the cast were not all from the same small set of drama schools. But it has now reverted and that kind of low budget, take a chance show doesn't get shown on the main channels.