It works both ways, no British actor or actress has been on screen using an American accent without me suspecting they're British. When I looked them up, I was right in all cases.
I think there are plenty of examples that could prove you otherwise. I can't say it's a case of me not suspecting they're English because I knew that they all were before watching the films/TV shows, but I easily forgot the following were English:
Idris Elba in The Wire
Kate Winslet in Titanic, Revolutiona2000ry Road, Mildred Pierce & Little Children
Domhnall Gleeson in Ex Machina
Rosamund Pike in Gone Girl
Carey Mulligan in Inside Llewyn Davis, Shame and Drive
Andrew Garfield in The Social Network
Imogen Poots in A Late Quartet
Rupert Friend in Homeland
Andrea Riseborough in Birdman and W.E.
Rebecca Hall in Vicky Cristina Barcelona, The Town and Lay the Favourite
Alan Cumming in The Good Wife
Kelly Macdonald in No Country For Old Men
David Oyelowo in Selma and A Most Violent Year. Hell, he even managed to pull off playing an English person masquerading as an American in The Paperboy.
There is one person who I did not realise was English until I looked it up and that was Olivia Cooke in Bates Motel. That is one flawless accent.
And Aussies do a good job:
Toni Collette in The Sixth Sense
Guy Pearce in Memento
Margot Robbie in The Wolf of Wall Street
Rose Byrne in Damages
And vice versa:
Renee Zellweger in Bridget Jones's Diary
Maggie Gyllenhaal as per mentioned
Michelle Williams in Incendiary
It's an ordinary high school day. Except that it's not.

@Gemma_Philips
Sienna Miller is English and does a fairly good American accent nowadays, a little 'predictable' (Factory Girl) but normalised in American Sniper IMO

You are right, the precise way overseas people speak English and usually in a slow way sounds unnatural to us, it's this unnatural way which leads to greater focus on it. From me anyway. You are correct about the dropping of consonants too. I have often pointed out to overseas people how English is perhaps spoken the worse by British people because of the consonant dropping. We also use idioms a lot which overseas people find irritating because you need to be a native to understand them, plus there are the euphemisms and our love of colloquial words.
I will listen to Maggie speaking again to see if she's dropping consonants and doing this convincingly enough to sound like a native. I'll add her to my list with Sienna if she does. I will give her extra kudos if she can throw in other aspects of our natural speech too - something Sienna does like a native. Of course being raised and educated in England helps Sienna and she does consider herself English.

So you mean a Posh English accent, something uneducated English people can do who were raised in slums.
Debatable, not to mention irrelevant as to whether Maggie Gyllenhaal's "posh" English accent was any good.
Sienna Miller is the only actress who I have watched and not detected something amiss.
Sienna Miller is American born but was raised and educated in England, so that's hardly surprising.
It's the yard stick If you spent the day with this actor speaking with an English accent, would you feel they were born and bred in England?
I was born and raised in England and I work with "posh" people within the legal arena, so yes, I would. Unlike other American actresses playing English women, I didn't detect one bum note, and that's in all eight episodes. Yes, she was sometimes precise and slow but that was perfect for her character, a woman with a controlled and demure public persona.
Americans can watch 'An Honourable Woman' and they will hear the American in Maggie like I hear the English with British actors.
Again, irrelevant. I'm English and her English accent was impeccable, even down to the tiny ommissions. A lot of American acctresses think that when you're doing a well-heeled English accent, you have to pronounce every consonant. Even posh people drop consonants and Maggie got this down to a tee. Watch the first episode when she's giving her speech: "Couple o' things" versus "couple of things". Brilliant.
I guess some people just have a better ear for it than others.
Well, there's your problem, see. Having only one ear, albeit a very good ear, will put you at a distinct disadvantage when judging the quality of accents, you poor luv.
I'm ****ing Irish. I'll deal with something being wrong for the rest of my life.

So you mean a Posh English accent, something uneducated English people can do who were raised in slums.
The best English accent I've ever heard from an American actress is that of Sienna Miller in 5b4Layer Cake.
Listening to Maggie most English people would know she is not British if they didn't already know this. Sienna Miller is the only actress who I have watched and not detected something amiss.
It's the yard stick If you spent the day with this actor speaking with an English accent, would you feel they were born and bred in England?
It works both ways, no British actor or actress has been on screen using an American accent without me suspecting they're British. When I looked them up, I was right in all cases.
Americans can watch 'An Honourable Woman' and they will hear the American in Maggie like I hear the English with British actors.
I guess some people just have a better ear for it than others.

The accent of someone who is raised and educated in England, and in Nessa Stein's case, the accent of someone raised and educated in the upper echelons of English society.
I'm ****ing Irish. I'll deal with something being wrong for the rest of my life.

Please explain what you class as a perfect English Accent?

I agree with you on the English accent. Boy, does she have a talent for utilizing that one even in the film Hysteria.