The Wire is very good.
It also features Idris Elba, so it has two British people playing Americans on either side of the law .
I also recall an episode of The Mentalist, where agent Rigsby (played by Welsh Owain Yeoman) says he can imitate a british accent and proceeds to do so, only to receive a “nah, not good enough” by the protagonist.
The issue I (as a non-native English speaker) sometimes have with British actors is: how to pronounce their names?! Examples: the aforementioned Owain Yeoman, Chiwetel Ejiofor, Siobhan (pronounced Shivohn IIUC) Finneran…
I don't think it's purely a non-native problem with those names, though they're not exactly English names.
I enjoy names such as Sean Bean ("Seen Been" or "Shorn Born"?) or even Mary Berry ("Merry Berry" or "Mary Bary").
[1] except David Tennant, who thinks https://www.youtube.com/watch?v=_gp9K-rMdxg sounds Californian
Probably helped by trying to get voice controlled technology working - particularly lifts.... ;-)
Wait... there are voice controlled elevators?
OK, so it's not just my wife; she sometimes directs voice actors, and within the first five minutes of the game as part of the big intro sequence everyone who plays the game has to see it got an "oof, that needed another take" out of her, LOL.