I know there are British actors who can sound American - like Christian Bale, for instance. Is the reverse true? Are there any Americans, either in the current day or in past times, who have successfully done British accents?
I don’t just mean who have done British accents, I mean who have actually done it to such a degree that a native British English speaker would not be able to tell that they’re faking the accent.
I have, in A Midsummer Night’s Dream. For four consecutive nights, I fooled audiences full of Brits. They were shocked (yes, shocked!) when they heard me speak in my normal voice: “What, are you American?!?”
Watch dialect coach Erik Singer critique actors’ accents in this YouTube series from Wired.
I quite liked Forrest Whitaker in The Crying Game
I seem to recall that most Brits praised Gwyneth Paltrow’s British accent in Shakespeare In Love and Sliding Doors. Also Renee Zellwegger’s in Bridget Jones’s Diary.
To this day most people I ask have no idea John Hillerman was American.
They probably never saw him in Blazing Saddles. 
James Marsters as Spike on BTVS has often been cited for doing a good Brit accent, which he credits to pointers from Tony Head (Giles).
I’ve heard it said that Spike’s accent is a copy of Tony Head’s actual accent (and not the Oxbridge he affected on the show).
It was bearable, compared to Cordelia’s, which was up there with Van Dyke with the worst ever english accents.
It’s ENGLISH accents, btw. British would include Northern Irish (such as James Nesbitt), welsh (I can’t name anyone using strong Welsh accent) and Scottish (countless but Ewan Macgregor (occasionally), Robert Carlyle (mostly) and, oh, yes Sean Connery).
Even within the remit of English accents there’s wide variations, with Newcastle, Yorkshire (half of Game of Thrones had that accent), Essex, Cockney, West Country, Manchester, Liverpool (the Beatles). The accent you probably should probably be called Oxford English, which is in effect what BBC presenters mostly use.
Cordelia didn’t have an English accent, did she?
Probably meant Drucilla, whose accent was indeed dreadful.
I may get flamed for this, but, of all people, Lindsay Lohan in The Parent Trap did four recognizably separate accents, and pulled it off quite well. As Annie, she had an American accent. As Annie pretending to be Hallie, an American putting on an English accent. As Hallie, she had an English accent, and as Hallie pretending to be Annie, an English girl putting on an American accent.
(I know this because my daughter was right in the target demographic when this film came out.)
Drusilla did have a bad accent. So did Angelus when he was supposed to be Irish.
The cast of This Is Spinal Tap all got critical praise for nailing a Squatney accent perfectly.
Squatney! What a delightful word.
Gillian Anderson has an excellent British accent in The Fall and Sex Education. Of course, though born in the midwest, she lived in England as a child.
She speaks with an American accent in the US, but an English accent in the UK. Here’s a comparison. I suspect she unconsciously switches depending on who she’s talking to.
In interviews, I find his AMERICAN accent unbelievable. I’m just that sucked in!
“I’m just a friend of Xandurrrrs…”
Agree Lindsay Lohan’s was very excellent in Parent Trap, possibly one of the best ever American pre-teen (or close) British accents. A bigger question would be why they are so rare- look at any American film with a large cast, probably half will be British or Australian, and you wouldn’t know until reading their imdb bios. There are many like Christian Bale and Will Poulter to name just two I would guess your average American film goer assumes are American- I know of at least two normal adults who were shocked to hear Batman speaking in a Welsh accent on a talk show.