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").