The most (in)famous example — cited for almost forty years now in anecdote, song and story — is Dick Van Dyke as Bert the chimney-sweep in Mary Poppins. The almost universal negative reaction to his overly fake Cockney was probably the reason that the next time he played an Englishman — Caractacus Potts in Chitty Chitty Bang Bang — he dispensed with the accent altogether. (Ironically, in Mary Poppins he also plays another Englishman, Mr. Dawes Senior, and disappears so completely into the role many viewers don't realize it's him until the final credits.)
This was actually a pragmatic decision. Dick van Dyke tried to do a realistic Cockney accent. And tried. And tried. And failed. And failed. Finally, he decided that since he could only do a bad Cockney accent, he'd do a hilariously bad one.
Notably averted in The Great Muppet Caper. A particularly reckless taxi driver comments about having lived in London his whole life. When asked why he doesn't have a British accent, he nonchalantly responds that he's lucky to even have his driver's license.
With Sean Astin bringing up the rear with his portrayal of Sam having a 'country' southern English accent.
Other Americans donning fake English accents of The Lord of the Rings: Brad Dourif as Gríma Wormtongue, who never used his normal accent in order to maintain it and only stopping when filming ended, which caused Bernard Hill (King Théoden) to wonder why he was suddenly using "such a fake American accent"; and Liv Tyler as Arwen, whose voice was so low that her own father wondered who the voice actor was. And also, Viggo Mortensen as Aragorn.
The movies were filmed in New Zealand and hence employed quite a bit of "local" talent. The numerous Aussies and Kiwis faking English accents include: Cate Blanchett, David Wenham, Karl Urban, Miranda Otto (with a hint of Irish) and John Noble. (Hugo Weaving grew up in both Australia and the UK),
Josh Hartnett's ludicrous attempt at a Yorkshire accent in the Keighley-set hairdressing comedy Blow Dry.
The Queen has a cast that's almost entirely British — except for Prince Philip, played by Los Angeles-born and Manhattan-raised James Cromwell. Prince Philip himself is a naturalized Briton having had to renounce his Greek ties before he was allowed to either marry the Queen (then the princess) or be served a peace-time Naval commission.
Alan Tudyk (of Plano, Texas) also plays a fake Brit in the British Death at a Funeral - both DVD commentaries (one with director Frank Oz and another with several cast members) complimented him on the excellence of his accent.
And poked fun at by English stand-up comedian Eddie Izzard: "I'm Raa-bin Hood! Where is the Maid Ma-ryan?"
Made all the more jarringly hilarious when he says "This is English courage" with a thickly American twang.
Claire Danes in Stardust pulls off an English accent quite well as Yvaine— particularly given that there's no reason that a star fallen from the sky should even have a British accent to begin with. Michelle Pfeiffer as Lamia... not so much.
Scarlett Johansson plays a British showgirl in The Prestige, a veritable treasure trove of fake accents. Oddly enough, her character was American in Christopher Priest's novel.
Given The Other Boleyn Girl portrayed events in English history and focused on an English king and his English love interests, you'd think they'd get British actors, at the very least, to play the lead roles (as opposed to Scarlett Johansson, Natalie Portman and Eric Bana). Not so, with the result that the vast proportion of English audiences spent the entire film wincing at the attempts the cast made at their accents (when they actually bothered to make an attempt, of course).
She also played Madeline Bray in Nicholas Nickleby.
And Emma in One Day, against actual Brit Jim Sturgess.
The made-for-TV movie Sherlock: Case of Evil cast New York native Vincent D'Onofrio as Holmes' nemesis Professor Moriarty with an accent that's just plain embarrassing.
American Michelle Williams sounded very natural in the part of Holly in film Me Without You playing opposite Anna Friel (who is English, from Greater Manchester) as they both play southerners (Received Pronunciation accent). In fact Williams pulls off the accent even better than Friel.
He also played the title role in Chaplin. Another character makes a snotty comment about his low-class accent, and he purposefully progresses to a more 'posh' tone as he gets more successful and famous. ("'Be'ah.' Betterrr. * snort* Bitch.")
The Piano has some of the worst Scottish (or Sco-tash) accents committed to celluloid. They range from 'reasonable' (Holly Hunter) to 'just plain bizarre' (Harvey Keitel, who accidentally sounds a bit Geordie at times). It was made in New Zealand with a largely Kiwi supporting cast but even so — it makes Groundskeeper Willie look like Robert Carlyle.
The version of Pride and Prejudice with Keira Knightley has varied accents — not surprisingly, veteran Canadian actor Donald Sutherland (renowned for his inability to do any accent but his own) does a dreadful job whilst young American actress Jena Malone (as Lydia) does reasonably well.
French actress Eva Green uses a particularly good English accent as the Bond girl Vesper opposite Daniel Craig's Bond in Casino Royale.
She's not doing an accent. That is how her voice really is now.
Chris Egan uses an accent somewhere between his native Australian and "posh" British in Letters to Juliet.
There's Something About Mary has an in-universe example. Tucker, Mary's 'English' architect "friend", is revealed to be an American pizza delivery boy named Norm. His actor, comedian Lee Evans, is British. So he's a Brit pretending to be an American pretending to be a Brit.
The Rocker has a scene at the end where after finally confronting Vesuvius, Fish realizes they all have British accents now. They all deny ever having been American. Later on in the scene the person who replaced Fish points out to him that he actually is British.
Nigel Tufnel and The Six-Fingered Man? Both played by the same American, Christopher Guest. Although, Guest's father was British - a hereditary peer, in fact, as he himself is now - which may have influenced him. He's not perfect though.
An in-universe example in Just Go With It, where Katherine's daughter is an aspiring actress who insists on speaking with a terrible English accent (the "'ello guv'nah" kind). They are forced to make up a story for Palmer where the daughter was in a British boarding school for a few years.
Not counting the Texan rat couple and the French Dragon frog villain, all of the characters from Flushed Away are voiced by predominantly British actors, except Roddy, who is voiced by Hugh Jackman (an Australian).
My Fair Lady contains the interesting idea of casting Audrey Hepburn as Eliza Doolittle - a character whose entire plot is based around her cockney-RP dialect shift. An interesting casting choice, that. Mind you, squire, she din't do an arf bad job.
While the woman singer dubbing Audrey as Eliza sounded acceptably English, the man dubbing Jeremy Brett as Freddy unfortunately sounded North American.
In-universe with the gentleman's club scene in Mystery Team.
Heather Graham in Miss Conception attempts received pronunciation and botches it horribly. Inexcusable really as they're in London and her co-stars are British.
Dead Man on Campus has an in-universe one with Matt Noonan, who gives off the impression of a suicidal goth rocker complete with British accent. When one of his friends finds out he is actually a happy-go-lucky guy who sings showtunes, his accent slips and his friend is shocked to find out he is American. He is played by Australian Corey Page.
American actress Julianne Moore's character in A Single Man. Apparently Moore listened to early Julie Christie for the sixties feel and mixed it up with modern British party girls to get her character's way of speaking.
Moore also did an excellent job of a 1940s upper middle class Englishwoman in “The End of the Affair”.
Selena Gomez in Monte Carlo as two separate characters. The first; her portrayal of the fictional English heiress Cordelia Winthrop-Scott and then her portrayal of an Identical Stranger from Texas, Grace, posing as Cordelia. Grace's fake accent is not entirely convincing, but that's kinda the point; Cordelia's accent is a bit better but that might just be because she had fewer lines and Gomez had fewer opportunities to mess up.
Americans Edward Norton and Liev Schreiber play nineteenth-century English gentlemen in the 2006 adaptation of The Painted Veil. The accents are pretty good. They at least avoid sticking out compared to Australian/Brit Naomi Watts.
Australian Chris Hemsworth as Thor, and Americans Jaimie Alexander as Sif and Josh Dallas as Fandrel in Thor put on pretty good English accents. They hold up quite well alongside actual Brits Tom Hiddleston and Ray Stevenson.
In Hugo - set in Paris, but everyone speaks with a British accent - American Chloe Moretz pulls this off impeccably.
Her effort at a Cockney accent was sporting and consistent, but laughable.
Prometheus stars Swede Noomi Rapace as the English Elizabeth Shaw. Her native accent trickles through quite a bit.
An In-Universe example in The Sting. Curly Jackson, a grifter from Baltimore who joins the con, likes to masquerade as an Englishman.
In the 1998 glam rock opus Velvet Goldmine, the Irish Jonathan Rhys-Meyers played a glittering bisexual English rock star (a David Bowie expy, in fact) rather well. On the flip side, the very Scottish Ewan McGregor faked it up really well as a big, bombastic, showy American rocker (an Iggy Pop expy) in the same film.
In the 2005 film The Best Man American Seth Green plays a Londoner. Those in the know say it's pretty much spot-on for a character of that particular demographic, only slipping up when he gets excited.
Straight and inverted in Lara Croft: Tomb Raider, where Americans Angelina Jolie and her dad John Voight play Brits, while Daniel Craig uses an American accent.
Tim from About Time is played by Irish actor Domhnall Gleeson who delivers a very convincing English accent.
South African Alice Krige specialized in this (except her most famous role as The Borg Queen).