This is an interesting one LanguageLover. The success of American actors trying a British accent ranges from the impressive to the truly awful! I could name several in the latter category - Including Dick Van ***'s legendary attempt at Cockney in "Meery Porpins" (Mary Poppins) and Mel Gibson was a bit dodgy with his Scottish accent in Braveheart. When Mel does an English accent he is more succesful - I think this is partly because of his Australian upbrining (Mel is pretty much "bilingual" in US/Australian English), and Australians seem to find an English accent easier to master because its not so different from their own. The worst "Americans doing British" usually appear in American sitcoms, where a US actor is used to play a Brit. Because they're often "one episode" characters, it seems they don't bother with accent coaches! So often, the actors attempt some kind of upper class, or "royal" accent (that less than 0.01% of Brits actually use), and yet miss the most obvious differences! (eg. the pronunciation of the "r" in words like "motor").
In the more succesful category, I would include Gwyneth Paltrow, as you suggest, in "Emma", "Shakespeare in Love", but especially "Sliding Doors", where she played a modern London girl. She sounded very natural to my ears. Another success was Renee Zellweger in the Bridget Jones movies - a bit of a departure from her Texan drawl! But even these are "safe" accents from the south-east/London area of the UK. What I would LOVE to hear, is a US/Canadian actor do a more extreme regional British accent, like Scouse (Liverpool) or Geordie (Newcastle upon Tyne), now that would be interesing! Does any one know any examples of N Am actors attempting these accents?