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TheSirfurryanimalWales Years ago, I was in a community theater production of "Dial M for Murder." (I was the police inspector.) The cast consisted entirely of Americans, and we worked on developing British accents (except for the guy who played the one American character in the show.) We were convincing enough for an American audience, but I had no delusions about how authentic we were.
I prepared for this by listening intently to BBC radio every night so I could imitate what I heard. I soon came to realize the very thing that you have just said: there are lots of different British accents, even within England. But I figured my job was to sound "as different from an American as possible" rather than trying to sound consistent .
So, in the course of delivering a single line, I probably moved from London to Liverpool to Cornwall. I did realize that that is what I would have sounded like, but who in the audience would know? (And if anyone came who really was knowledgeable about British accents, nothing I tried to do would have been convincing anyway!)
Cheerio!