Why is “bloody” considered obscene in the UK but not 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.