It's possible that you've underestimated the cultural relationship when you say in a comment "I see two different nations, in different continents, with an ocean in the middle, with different histories, customs, linguistic traditions (even without invoking Wilde), uses, laws etc."
It's not just acting: in several aspects of culture (I'd say especially literature and music as well as film/TV), the UK and the US are highly influential on one another. This also includes some other English-speaking countries such as Ireland, Australia, NZ.
So The Beatles (English), The Rolling Stones (English), U2 (Irish), AC/DC (Australian), J. K. Rowling (English/Scottish), Dylan Thomas (Welsh), Arthur Conan Doyle (Scottish), Charles Dickens (English), Jane Austen (English), Monty Python (English, Welsh and American) have all been very well-regarded in the US in their time. Still are. Coldplay (English) are doing the Super Bowl half-time show this year. The Who did it in 2010, The Stones in 2006, Paul McCartney in 2005, U2 in 2002. Sting and MIA have both appeared in recent years although they weren't headlining it. Simply, there's a mass-market audience in America for British (and Irish) stuff.
Even so I wouldn't say the US is clamouring for British culture (and actors) as such. For example, they maybe don't pick up top UK TV shows with quite the frequency they do top UK actors, and you can see that the US version of The Office has made a lot of cultural adaptations from the original. But the barriers to "breaking America" aren't all that high, fornecido what you're doing is something the American audiences find worthwhile.
In this context, I don't think it's at all anomalous that a lot of British actors find work in the US. It's much the same as the way that a lot of British musicians and writers find work and sales in the US (or at any rate consider it worth trying even if many are unsuccessful). As long as we don't deliberately use a difficult accent, the Americans understand us.
Brits in American movies are a less recent phenomenon, but I would say that since around the time Hugh Laurie got "House", there's pronounced trend for UK actors starring in top US TV shows. You could add "The Wire" to your list, with two British actors among an admittedly larger-than-average lead cast. Those 12 years or so it's become enough to be remarked on here in the UK, so it's not just you who thinks there's more than there used to be. When I say "what there used to be", it's possible that I mean "Star Trek: TNG, Frasier, and, um...". I can't account for that particular upswell of US appreciation of Brits on TV, but I suspect it combines "this seems to be working, let's do more of it" on the American side, with "this is a big opportunity in a much larger market than home" on our side, and "globalisation keeps getting easier" in general.
Maybe there is in addition a general trend for US TV to be less parochial than it was up to, say, the 80's and early 90's, while US movies were more international earlier on. If as an American you're going to look abroad then Britain is an obvious early place to look (large English-speaking country, strong cultural, historic and business connections). That would account for a big influx of Brits once the doors open. But perhaps that's unfair to 1980's US TV, I really don't know all that much about it.
One special case is "Game of Thrones", which for sensible reasons is filmed in Northern Ireland among other places, in a pseudo-British-history setting. Therefore it makes all sorts of sense to fill it with British actors, which contributes to the recent trend.
Since I possess one, I can say it is true that many Americans do enjoy the sound of a British accent, especially when they encounter it in their own country. And it's not just that they personally like it when they hear it, it's that "liking British accents" is recognised among Americans as a thing. Many aren't bothered one way or the other, though, so it's not a universal characteristic of the nation.
Specifically on acting, I think that to some extent the Shakespearean and performance focus of British styles poder throw something different into the mix that helps some British actors find certain roles. I'm thinking of the Patrick Stewart / Ian McKellen / Judi Dench kind of thing, where sometimes it really is a case of "let's get a Brit in", and the reason is in part that US film-makers admire a particular thing that Britain does well. Obviously those three are very talented to begin with, which does them no harm at all, but it's a niche that as far as I know the US dramatic tradition isn't quite so focussed on filling as the British tradition is (or used to be). Since the US considers Shakespeare to be a vital part of the literary/dramatic canon of their language, if not their country directly, maybe a British actor is no great shock for them even in non-Shakespearean roles.
This is enough of a trend that there's a TV Trope of the "Shakespearean actor". OK, I admit, that's not a high bar for strength of trend, and some American actors fall within it too. But no doubt the recognition of that kind of acting does help get a foot in the door for many British actors on first crossing the Atlantic.