In GB, it seems that everyone takes a turn on Midsomer Murders, New Tricks, The Doctors, Coronation Street, Doc Martin, etc. I think that's because they're not locked in to 26 episodes per season like in the US.
Many moons ago, the British programs I watched were
Doctor Who,
Monty Python's Flying Circus and
The Hitchhiker's Guide to the Galaxy. Not much overlap there, though Peter Davison appeared in
HHGG as a talking cow that wanted to be eaten and John Cleese was also in an episode of
Doctor Who. (And then there's Douglas Adams who was associated with all three, thought not necessarily on-screen.)
But, anyway, largely not recognizing actors meant it was a little easier to fall into the stories because then you just see the characters and aren't thinking about it being just actors in a role.
Plus, the lack of familiarity with British politics was helpful too for those occasions when a character is a politician. Basically, anyone other than the Queen or Prime Minister is pretty much unrecognizable. So, if you say this character is the whatever of whatever, it's like sure because you don't know any differently. Makes the story feel more real. Whereas watching American TV, and you have a character that's, say, the Senator from such and such, you're more likely to be like, wait, no, it's not. And that can take you out of the story a bit.
Shows like
Stargate: SG-1 avoided that by referencing the president but not by name, so you could imagine they were talking about Clinton or Bush instead of some fictional president. And that made it seem more real that, you know, maybe there is a secret Stargate program in Cheyenne Mountain. One that's not a janitorial closet. Of course, then they had Senator Kinsey and later had a fictional president but they did balance it out a bit by having, for example, the actual (at the time) Air Force Chiefs of Staff appear.