I can’t think of anyone with a more perfect voice for reading aloud from ‘Fifty Shades of Grey’ . On youtube. When I desperately need a laugh, I watch that and it’s like magic. If he only actually read the whole book, I would so buy that!
Big Arnie has claimed in interviews he can speak American English without any accent but he deliberately adds some because his fans expect it. I don’t exactly follow his career closely but I cannot recall ever hearing him speak without some.
However there is a certain irony that apparently when he speaks in German these days that now has an American inflection.
TCMF-2L
That actor with the swoopy eyebrows. I just saw him in the latest Black Mirror episode Bandersnatch and thought “his British accent is pretty good!” He is British - I’d only heard his American accent.
Fran Drescher sounds the same as The Nanny and as herself, so I expected Megan Mulallay to sound the same as her Karen character from Will & Grace but nope, she sounds normal. You do get a sense of the Karen voice from her speaking but she’s doing a big voice for the show. Same as the aforementioned Melissa Rauch.
I find that hard to believe since back when he was still governor there was a joke that he was the only California governor who couldn’t pronounce California.
“Cully-phone-ya.” 
I had no idea of Joan Plowright’s origin when she played Tracey Ullman’s Yugoslavian mother in I Love You to Death. I thought she was a genuine South Slavic, but she’s British.
I believe it depends largely on where she is living at the time. When she’s in the States, her American accent comes out, and when she’s in England, her English accent is prominent.
John Barrowman is the same way; he was born in Glasgow but moved to the US when he was a child. I’ve seen him on American talk shows speaking with a typical American accent, but when he’s in Britain the Scottish really comes out.
My first exposure to both Dominic West and Idris Elba was from The Wire. I was quite surprised the first time I heard both of them using their natural voices.
Me too, ditto Aidan Gillen.
When asked if that’s her real voice, Fran Drescher’s standard reply is “Please. Who would make this up?” She does know how to speak in a “normal” voice.
The person I think of is actor Paul Sorvino. His typical roles include authority figures on both sides of the law and even his comic roles (I loved him as Bill’s father in the sitcom Still Standing, which is one of my all time fav shows) tend to be gruff, authoritarian guys.
The first time I heard him interviewed I was surprised and a little delighted, because he came off as really suave, sophisticated and cultured, kind of the polar opposite of the cop/gangster role is was known for.
Semi-related: it seems that one easy way to spot a guy for whom English is not their first language is to ask them to say “squirrel”. Apparently it’s really hard for Germans.
Check out Peter Jackson’s Heavenly Creatures.
Do play-by-play baseball announcers count as actors? No? Well, if they did, I’d nominate Howie Rose, who does mostly radio for the New York Mets. I heard him on a morning radio show one day after a game, and had no idea who he was before being told. His voice was an octave lower (okay, not that much, but still) and much slower. He definitely amps it up on game day.
When I heard James McAvoy interviewed on the Graham Norton Show I almost needed subtitles to understand his Scottish accent.
Mike Tyson
(Was in “The Hangover”, so qualifies.)