As an admirer of Shaw, I was impressed that the successful musical version of Pygmalion, "My Fair Lady" was created by Americans (and acted by Brits).

In the early days of TV and television shows through the 70s, in my experience, it was more likely to see American actors in English movies and shows acting other than American. They could convince Americans they were acting English, I don't know how convincing they were to the English. Over the past twenty years many English actors have done believable American accents and characters, especially Hugh Laurie, Damian Lewis, Idris Elba, and many female actors as well.

There is also a fun and well done American movie about voice actors and accents: "In a World".