In my experience, the biggest sin of British actors trying to do American accents is that their “American” accents often tend to be pretty New-York-ish, whether or not that is appropriate to the character. Dominic West as McNulty in The Wire is an example of this. He sounded more “New York” than “Baltimore.”
The accents of the stereotypical fat American tourists in In Bruges are pretty awful.
And your definition of “grew up” is exaggerated. She was in England basically from 2 to 11. Then she lived in The U.S. From 11 into adulthood. So by the time she was in the X-Files she had lived in the U.S. 16 out of 25 years.
I wouldn’t say exaggerated exactly. Maybe he is trying a bit too much to have a neutral American newsman accent. But to me it doesn’t sound unnatural.
Yeah I never thought West’s accent was great. Not the worst but not exactly natural. Unlike Idris Elba on The Wire. I never would have thought he was not American. Or Eamon Walker on Oz. Perfect.
No. But I remember LL having an accent that wasn’t particularly easy to place, but not quite right for “generic American”, and nothing notable about Renee O’Connor’s accent at all. Certainly didn’t sound very Texan to me (not that I could likely accurately place the various south western accents anyway), nor was it in any way hard to understand. If either of them sounded “generic American” it was Renee.
For me, the worst “American” accent by a non-American actor was in some show called Demons they ran briefly on BBCA. Philip Glenister’s accent on the show was so non-American, it makes me wonder if he’s ever even met an American or listened to an American speak in his life. Also, his character was named Rupert… How many Americans are named Rupert? I’m sure there may be an outlier or two, but it has to be approaching zero. The character was quite explicitly supposed to be American, though.
Heh, “Ihm uh frennd of Xandurr’s”
Archie Panjabi who plays Kalinda on The Good Wife drives me crazy… She’s supposed to be American, but constantly loses her accent and sounds British…
This is my answer as well. I had actually seen quite several of the stories featuring Peri before Rhiannon8404 informed me that she was supposed to be American.
My viewing is so split between American and British shows, plus calling all over the English-speaking world for work, that I’ve mostly stopped noticing that anybody has an accent.  Plus people’s accents are getting homogenized.  
Damn. Quintuplely ninja’d. That’s a personal record.
Philip Glenister (the much-loved DCI Gene Hunt in Life on Mars and Ashes to Ashes) played an American of some kind in Demons. The accent was diabolically bad.
I’m surprised no-one has picked up on this, because just the thought of Ray Winstone trying to battle the unstoppable force of his east end brogue made me grimace.
I couldn’t actually find a clip from that movie, but in this Adam & Joe youTube audio file, they play various clips of him trying to do an American accent (note: there’s a long intro. The first example is at 1:50)
No, he’s saying that her accent when being Alison imitating Sara (or Helena imitating (I think) Beth) is hilariously (intentionally) bad. And it was in Dick Van Dyke territory. Which is great acting.
I know, right? I was sooo shocked when I heard his natural accent.
No one has picked up on this because:
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It is a thread killer. The question: The worst American accent? The answer: Ray Winstone in The Devil’s Tomb. The children can squabble over Gillian Anderson all they like. The Daddy has spoken. The OP has their answer. 
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Probably no one else has actually watched The Devil’s Tomb. It is a horror film and is, appropriately, quite gory. It has high (-ish) production values. But the script and the acting are dreadful. 
As for Ray he only has a cameo in the film (and his role adds very little to the narrative) so he doesn’t appear in the trailer. He pops up regularly in flashbacks although he only really speaks in an extended appearance towards the very end of the film.
TCMF-2L
Yeah, I totally misread that. In my defence, I totally misread a lot of things.
Re: Gillian Anderson. Before I knew any of the above tidbits, I just assumed she was Canadian. I guess that’s my go-to when a U.S. accent seems a little “off.” Plus, I think most of the show was filmed in B.C. or thereabouts.
Part of that is my own experience. I’ve lived in several regions of the U.S., including Chicago, Arizona, Houston, and rural Georgia, plus I lived with people off and on from Eastern Europe and England, so (esp. as a teenager) people would try to guess my accent. Inevitably someone would say I must be from Canada, and other people would say, “Oh, yeah, duh.” LOL. Apparently I enunciated too crisply for everyone, but drawled the wrong things, too. But I am the worst person for unconsciously mimicking accents. I imagine Anderson has a similar issue.
As such, the other actors mentioned above–those I’m familiar with–I don’t really see issue with. Peri from Doctor Who sounded pretty Midwestern to me (had to look up some clips, AncientHumanoid…she was distractingly cute). The problem is, the writers still gave her plenty of British constructs in her speech that she had to work around.
People make fun of Andrew Lincoln, but he also doesn’t sound too off to me.  I never, ever, would have guessed that he was English.  Maybe someone from Kentucky trying to sound more Deep South, but that’s about it.  For fun:
Accent Tag - Atlanta, Georgia - Authentic Atlanta Native Accent - YouTube (some lady from Georgia)
Andrew Lincoln on Developing Rick's Accent for The Walking Dead - YouTube (Lincoln talking about his accent shift)
Hugh Laurie sounds natural to me, also. In fact, his “House” accent sounds like a lot like me, except that he doesn’t round his vowels enough. And the gender disparity, of course.
I know people from the part of New Jersey where House’s hospital is supposed to be, and they talk like that. I really think Hugh Laurie must have been imitating a particular person, his accent was so good.
I watched a little of Gracepoint to see the fuss about David Tenant. My husband said he thought the accent was OK, but weird, coming from Tenant. I’m not a Doctor Who watcher, so that meant nothing to me, but I thought his vowels were fine, he didn’t trill his Rs, and so forth, but there was still something about his manner of speaking that wasn’t American. Maybe it was his intonation, or maybe his character just didn’t come across as an American cop. I should watch more of the show, probably, I just watched about 30 minutes of clips. Still, it seemed to me that if he’d been playing something else, like the headmaster of a private school in Boston, or a district attorney, or something, he could have pulled off the accent, but there was something all wrong, almost uncanny valley about him as an American in that particular show, in that role.
Madonna apparently lives somewhere in the middle of the Atlantic Ocean.
Years 1.5 - 11. The ages you actually learn to speak. Her British accent is her natural accent. The American one was acquired later.
:smack: I must have a terrible ear because I thought Hugh Laurie did fine on House (except for maybe an off word very infrequently) and David Tennant did fine on Gracepoint.
Once you do know, though, you can hear the English accent leaking out all over the place. He struggled with the R’s sometimes. Like how he pronounces ‘work’ here: https://www.youtube.com/watch?v=gGgRtiCVo2w
Also how he says ‘do it.’