Why is the word bloody considered offensive in Britain — where it is used as an adjectival expletive — but not so in the US?
I don't think it's obscene so much as profane. By one theory, "Bloody" in this context is a contraction for "by our lady", essentially swearing by the Virgin Mary (Bloody in Wikipedia). Other similar oaths include "blimey" (God blind me) and "gadzooks" (by God's hooks (hands)). As to why "Bloody" is considered obscene/profane in the UK more than in the US, I think that's a reflection of a stronger Catholic presence, historically, in the UK than in the US, if we're accepting the above etymology, as Catholics venerate the Virgin to a greater extent than Protestants.
All of which is void, of course, if the etymology is incorrect.
Here’s the OED’s comment on the origins:
The origin is not quite certain; but there is good reason to think that it was at first a reference to the habits of the ‘bloods’ or aristocratic rowdies of the end of the 17th and beginning of the 18th c. The phrase ‘bloody drunk’ was apparently = ‘as drunk as a blood’ (cf. ‘as drunk as a lord’); thence it was extended to kindred expressions, and at length to others; probably, in later times, its associations with bloodshed and murder (cf. a bloody battle, a bloody butcher) have recommended it to the rough classes as a word that appeals to their imagination.
I always thought "bloody" was a reference to the blood of christ. kind of like "suffering Jesus" if you've ever heard that.
edit: I see that this interpretation is also listed on Wikipedia, nonetheless, it uses something of religious value and turns it into an expletive. I think it's more offensive in the UK than the US because it started in the UK, and people actually know what it means, whereas in the US "bloody" is just a funny British curse word. Don't know if that helps in any way.
Comparing UK usage...
...to US usage, I think it's quite clear that US usage lags behind the UK here. Bear in mind that sheer weight of numbers normally puts the US ahead, so any suggestion of the opposite trend should probably be seen as more pronounced (and significant) than the charts might suggest.
Also compare prevalence values in British and American corpuses for stupid bloody, to see this more obviously "intensifier" usage is almost three times more common in the UK than in the US. For example, I hear things like "You bloody idiot!" all the time in both real life and British movies, but in American movies (and, I suppose, real life) it's much more likely to be "You fucking idiot!".
I think what this means is that many middle-aged and older Americans (the age range most likely to be defining any usage as "obscene") barely even encountered adjectival bloody when they were growing up. So they had no predisposition to rail against it when the younger generation started using it more. The etymology of the word (which has been addressed repeatedly on ELU) is at best uncertain, but most likely few Americans automatically perceive it as blasphemous/profane. So to many of them, it's just another blinking, blooming, freaking, flaming, flipping mild "cuss word".
Brits, on the other hand, were using it a generation earlier - back in the days when using even relatively mild curse words was generally considered a "bad thing".