A fairly recent episode of Blindspot made me think of it. How many examples are there of an actor using an accent other than their own for a role and then as the character pretending to speak in the actors real accent? Either for comedic effect or dramatically.
In Blindspot the character Roman kills an Australian and takes over his life. Roman has been on the show for more than a season and has an American accent. The actor Luke Mitchell is Australian. There is a scene where he is trying out his new Australian accent in the mirror.
Hugh Laurie once used a very bad English accent as House.
In the 1st season of The Wire Dominic West does a poor English accent while undercover.
I think this is an example: In There’s Something About Mary, Lee Evans, who’s a Brit, plays an American who’s pretending to be British. Not sure if he uses his true accent, though.
In The Paperboy (2012), Brit David Oyelowo plays and American who is pretending to be English so as to avoid racism in the American south.
Not sure if those are the kind of thing the OP is looking for.
In The Island, Ewan McGregor plays a clone of a famous celebrity - when the actual celebrity character shows up, Ewan McGregor uses his normal Scottish accent.
In the American show Chuck, Australian actor Yvonne Strahovski played an American CIA agent. In one episode she pretended to be an Australian researcher of some sort, complete with strong accent.
Not really what the OP is looking for but I had seen Margot Robbie in several movies playing American characters and I assumed she was American. I then saw her in Whiskey Tango Foxtrot and I remember thinking she did a good job on her character’s Australian accent. It was a few months later that I read she actually is Australian.
I once heard Gary Oldman on Wait, Wait, Don’t Tell Me say that he has done such a wide variety of roles that he is no longer sure what his normal accent is.
British actor Franz Drameh plays an American on Legends of Tomorrow. One episode he went undercover at NASA as an observer from the British space program.
On White Collar, Agent Berrigan had to “pretend” to be English while undercover, giving actress Marsha Thomason a chance to bust out her native Mancunian.
In “The Agency”, Irish actor Jason O’Mara played an FBI agent. In one episode, he went under cover as an IRA operative. People complained about his horrible Irish accent.
In the movie The War of The Worlds Alan Young appears as his own descendant, he uses his original Scottish accent and his descendant uses an American accent.
I know this is really the opposite of what your talking about I thought it was pretty funny where in one episode of DuckTales Scrooge losses his accent.
Yes Time Machine as was mentioned. But that’s not his accent. If you listen to interviews he had a Canadian/American accent with a hint of Scottish on some words. He only lived in North England and Scotland until his parents moved to Canada when he was 6.
He spent much of his career playing Englishmen, most notably in “Magnum P.I.” But he was actually a Texan. In one “Magnum” episode, he played Higgins’ cousin, a rodeo cowboy, and got to use his natural drawl.
Dan Stevens encounters an imaginary British-accented version of himself in Legion. Stevens is of course British.
ETA: Regarding the Gary Oldman thing, Patrick Stewart has also commented that he’s long since lost his original Oldham accent and now can only fake it.