At one point while interviewing Ricky Gervais, Conan O'Brien decided to do a bit in a fake English accent. After asking if it was supposed to sound British, Gervais comments "Thank you Dick Van-fucking-Dyke!"
The three-part episode of the Adam West television series where they end up in London, sorry, Londonium, for a while featured so many atrociously bad English accents and fundamentally stupid errors (Ireland Yard?) that one can only hope it was intended as a parody.
Spike, Drusilla, and Wesley. Spike's accent is said to be based on Anthony Stewart Head's regular voice, which is a lot less posh than Head's character Giles. Yet Spike is shown in flashback to originally have an upper class English accent, so whatever he's trying now is an affectation.
Like Dominic West below, at one point, Spike puts on a American accent, which is hilariously bad.
Alexis Denisof (Wesley), a Maryland native who lived in the UK for much of his early career, is the Buffyverse's most convincing fake Brit; even British fans don't always realise he's not English as long as he doesn't use the typical trip words such as 'data' where it becomes painfully obvious even to Britons who were previously fooled. Drusilla, on the other hand, spends a lot of time talking about "Spoik". In fact, Juliet Landau sounds like she's doing a particularly bad impersonation of Harry H. Corbett.
Also, both of the "English" potential slayers, one with a bad Cockney similar to Drusilla, and another with a bad RP accent. God knows how genuine Brit Anthony Stewart Head could put up with filming with them.
He answered that very question in an interview: "Money soothes all ills."
Everyone from Spike's past in "Fool for Love". It might be easier to list the characters in Buffy who were played by actual Brits.
Buffy as well as the spinoff Angel had a bad case with one of the Watcher's Council special ops team. While their leader was played by real Brit Alistair Duncan (a Scot who now has a more English accent), Jeff Ricketts played one of his underlings, at which point the word "vampire" becomes "vampoire" and the only words he seems to get right are curse words such as bastard.
Castle: Colin Hunt from the fourth season purports to be from Scotland Yard. The actor's actual Australian accent is painfully obvious in his attempts to do some sort of mangled Cockney.
CSI: Used and an Inverted Trope in one episode in which a man plays Sherlock Holmes. When his friends, who were invited, first show up, they all speak in English accents. When they realize their host has died, they drop their accents, except for one — who turns out to be English.
CSI NY: Claire Forlani is British, but even so, Peyton's accent was made some sort of Fake Brit accent.
The Dresden Files: The TV series featured Fake Brit Terrence Mann (who was born in Ashland, Kentucky and who grew up in Largo, Florida) as ghost-with-a-Teutonic-name-and-a-British-accent Hrothbert of Bainbridge. Bainbridge, by the way, is a real town in North Yorkshire.
Extreme Makeover Home Edition: Sometimes even Brits fake a 'British accent' for an American audience: for a while on this show one of the team was a sort of cheeky-chappy jack-of-all-trades, whose Dick-Van-Dyke-alike hamming up of a Cockney accent (as well as acting up to other cockney stereotypes in a 'why fank you guvnor, i am ever so 'umble, cheerio, lawks etc' way) made it hilarious to realise he was genuinely English.
The majority of the cast are Australian, with the obvious exception of Ben Browder. A few Peacekeepers keep their accent, but the majority of the regular characters disguise it with either Fake American or Fake Brit, most notably Aeryn, Scorpius and Crais.
Oddly enough, Claudia Black's Australian accent is so close to British that Aeryn uses her natural voice.
Black also notes in an early interview that other people playing Peacekeepers weren't entirely sure what type of accent to use since Black's odd conglomeration of Australian and British was their baseline. This is probably why the end result is a wide gamut of native Australian to faux-Brit, with the occasional faux-American.
And when John Crichton impersonates a Peacekeeper in one episode, he puts on a not-great English accent.
In the episode Shindig, River Tam mimics Badger's authentic London accent.
The episode commentary from Morena Baccarin and Jane Espenson contains some fine Fake Brit moments. Both figures praise Summer Glau for her ability to do accents very well (unfortunately, most British people would disagree on her supposed London accent). Espenson also goes on to mention her love of British slang words and how she likes to use them in her work. The example used is the word 'palaver' - meaning 'an unnecessary fuss'. She pronounces it erroneously as 'par-layver' (it's 'puh-lah-ver' (or 'puh-lah-va' if you're from the London area).
While Daphne Moon was played by British Jane Leeves, her brothers Simon and Stephen were played by Australian Anthony LaPaglia and Swazi Richard E. Grant (his accent is clear RP, however). While Leeves' 'Mancunian' was a rather generic Oop North accent, the brothers didn't even sound like they were from anywhere near there.
None of the Moon brothers with speaking parts — or her father — were played by Englishmen.
Any of Daphne's 'chim-chimerny' boyfriends from early series - clearly played by American actors who think saying 'cheerio' makes them English. Embarrassing.
During one episode where Ross has to lecture in NYU, he gets so nervous he starts speaking in an (appalling) English accent. Monica and Rachel mock him by speaking in terrible Irish and Indian accents, respectively. Monica's seems to fluctuate between Ireland and Scotland, possibly as a reference to this trope or just because Americans can barely tell the difference.
Also spoofed in an episode where an annoying old friend of Monica and Phoebe's comes back into town after living in England with a fake accent:
Amanda:(after an awkward comment) Oh! Bugger. Should I not have said that? I feel like a perfect arrrse!
Phoebe: Yeah, well, in America you're just an ass.
Game of Thrones: Although his character isn't technically British, Peter Dinklage adopts a cod-English accent to fit in with the almost exclusively British cast of this show.
Gossip Girl: American actor Patrick Heusinger played Blair's boyfriend Marcus, an English lord, in season two. Extra twisty points for the fact that Marcus pretended to be an American throughout most of his first episode. Later on his English accent was copied by Chuck, who wanted Blair to think he was Marcus. Chuck, an American, is played by a British actor...
Highlander: Duncan MacLeod. Adrian Paul used a Scottish accent in the flashbacks of Duncan's early years, but made it more generalized in the more recent flashbacks.
Lampshaded, where the Jerseyite Dr. Jerk played by a Brit calls a hospital several times trying to get info. At one point, he uses an English accent, to which the operator responds "And that's the worst English accent I've ever heard!" Brilliantly done by Laurie, as he wasn't using his native English accent, but was doing an American's-poor-attempt-at-English-accent.
The Inspector Lynley Mysteries: DS Barbara Havers is working-class English. She is played by the very Scottish Sharon Small. Fans who first encounter Small as Havers are often stunned when they hear just how wildly her native Glaswegian accent differs from Havers' Estuary English.
K9: This show is set in London. It's filmed in Australia. The accents vary, but some of them are terrible.
Law & Order: Criminal Intent: On one episode, detectives Goren and Eames are unraveling a murder and encounter a "British Lord" who does not realize he is part of a scam (he's an actor who thinks he's been hire for some kind of performance piece) and spend a few minutes trying to figure out where his accent is supposed to be from. He drops the accent and attempts to impress them with his acting bona fides.
Averted Trope as far as the actors are concerned. Gina Bellman (Sophie) was born in New Zealand to English parents and moved back to the UK when she was eleven, so her accent is genuine, as is Mark Sheppard's. However, the characters sometimes have to put on fake accents for a con, with Sophie making hers more downtown London than Upper-Class British Thief for "The Beantown Bailout Job", and Hardison taking on a Londoner accent in "The Ice Man Job".
Hardison's "Londoner" accent is painfully bad, but justified since he's not normally the grifter.
LOST: Alan Dale, who is from New Zealand, plays British character Charles Widmore. In his first few appearances the accent was impeccable, but his accent slipped a little in "There's No Place Like Home." He also played King Arthur for a time in the West End production of Spamalot.
Magnum, P.I.: John Hillerman, a native Texan, played English ex-military vet Jonathan Higgins. During the show's run, his character was required to "fake" a Texas accent to impersonate his look-alike half-brother Elmo. (Like Niles in The Nanny and Wesley Wyndham-Price above, a lot of British viewers were convinced he was a real Brit.)
Amongst the real Brits, Colin Morgan of Merlin is Northern Irish and his real accent shows it. He used a British one for the series.
The Nanny: Niles. A story goes that viewers of this show in the UK wrote in to complain about the "fake" accent used by Charles Shaughnessy (a real Londoner) and praising the "real" accent of Daniel Davis (Niles), a native of Arkansas.
Orphan Black: Tatiana Maslany and Jordan Gavaris, both Canadian, play British orphans who immigrated to Canada. In Maslany's case, she's Acting for Two, so it makes sense (at least from a real-world perspective) to adopt a fake accent to differentiate her characters from each other.
Penny Dreadful: American Reeve Carney's British accent as Dorian Gray can be quite shaky at times.
Rumpole of the Bailey: Australian Leo McKern spent his career playing Englishmen, most notably Horace Rumpole.
Sanctuary: Amanda Tapping, while she was born in England (Rochford, Essex to be precise) has lived in Canada since she was three. Lampshaded in "Bank Job" when she reverts to her Canadian accent and the bank teller (Gary Davies who plays Sgt. Harriman in Stargate SG-1) comments "I knew that British accent was fake".
Seinfeld: Jerry tries on a Cockney accent, where "Not bloody likely!" comes out as "Nawwt blooudy loiklay!" Kramer criticizes it, but his is no better.
Star Trek: The Original Series: James Doohan, first generation Canadian-Irish descent, as Montgomery Scott in this show. His audition for Star Trek was pretty much him speaking in as many accents as possible. He chose to have the character be Scottish, because "all the best engineers are Scottish". Also, his talent with accents caused him to voice nearly every male guest character in the Animated Series. He ended up being a Fake Native American, among others.
Strike Back: American actor Philip Winchester portrays English Sgt. Michael Stonebridge on this show. Doubly ironic as Stonebridge's counterpart Damien Scott is a Fake American.
The two that immediately spring to mind are Jonathan Rhys Meyers as Henry VIII and Sarah Bolger as Princess/Lady Mary Tudor. Both are Irish but are playing Brits. Pretty sure there's others as well, since the show is filmed in Ireland.
Princess Elizabeth is played (as of season 4) by Dublin born actress Laoise Murray, meaning that most of the actual Tudors on the show are played by Irish actors. No information on whether Eoin Murtagh (Prince Edward) is Irish or not but that is a pretty Irish first name.
The Vampire Diaries: The Originals have British accents despite being Scandinavian. Klaus is played by real Brit Joseph Morgan and Rebekah is played by Australian Claire Holt. The Originals all have different accents. Most notably Klaus has a Welsh accent, Kol has a London accent, Elijah has an American accent and Rebekah has an Australian accent.