Commenter Matt McIrvin thinks things generally have improved:
There used to be a lot of British actors doing terrible American accents. You can hear them if you watch old sitcoms and Doctor Who serials from the sixties and seventies (the rocket pilot in “Tomb of the Cybermen” is a particularly choice example). Most of them seemed to be doing one of two accents. One was an outrageously exaggerated cowboy drawl used mostly for comic effect; but the other, used for non-cowboys, sounded sort of like an attempt to imitate the style of 1940s radio or newsreel narration—this rapid, somewhat nasal, loud barking patter that I’ve never heard any American use in actual conversation. I assume that acting classes were actually teaching it as a generic American accent.
Used to? For all my love of Inspector Morse, I don’t think any of the alleged “Americans” on that series ever managed to sound like an American—most sounded like a British person trying to imitate John Wayne and failing miserably because they weren’t John Wayne and no other American talks like him. More recently, the less said about the accents of any of the actors in Dalek during the first season of the Doctor Who revival, or the president they killed off last year, the better.
John Barrowman sounds like an American on Torchwood, but that’s largely because he is one. (The incongruity there is that nobody ever seems to be very bothered by the fact he’s running around Cardiff dressed like Bomber Harris and talking like an American, not even the people who ought to be predisposed to dislike Torchwood. Or Americans, for that matter, which may be an overlapping set.)