You can pick-up some very useful dialect tips from this recent 8-minute BBC interview, whether you are an American trying to do a British accent, or a British actor who is trying to sound American.
Dialect and voice coach, Elspeth Morrison, reviews and critiques several actors' recent performances, including Dakota Fanning, Emma Watson, Dominic West and Damian Lewis, and analyzes what they are doing right and what, if anything, they are doing wrong. (Spoiler: She is pretty complementary and positive overall.)
BBC Radio: "Description of this segment from the Front Row web page: Recent screen roles for British actors Damian Lewis and Emma Watson have demanded American accents, and American actors including Dakota Fanning and Maggie Gyllenhall have recently used British accents for parts set in the UK. Voice coach Elspeth Morrison discusses the techniques needed for these transatlantic accent swaps."