I use expecting quite a bit, esp. when talking with co-workers or acquaintances. It also has the advantage of allowing one to use the 1st person plural ("we're expecting"), which doesn't work with pregnant ("we're" not pregnant, since only one of us can be).
It also allows one to continue on to say the due date, e.g. "We're expecting a baby around June 20th," rather than how far "along" one is, e.g. "six months pregnant." The second instance hearkens back to the act of conception (and I think you're right about most people not wanting to point out the obvious biological aspect), while the first allows one to look forward to the reason you're talking about the whole thing in the first place—a new person is going to be around.
It may not be British in origin; it just seems to me that I wasn't really aware of the term until I started watching British TV on a regular basis. I imagine a good dictionary of slang could resolve the question.
As to why people use it, that's interesting. What you call someone who is pregnant is sort of in the same class as how you describe a couple who live together without "benefit" of official matrimony, and what they call each other. We don't have a simple, casual word for it that doesn't point to the (largely illusory) awkwardness of the situation. The word pregnant seems OK to me, but it does carry a whiff of the biology textbook, complete with line drawings of the associated reproductive organs. Words evoke pictures, and though in most cases pregnancy is of course something we celebrate, people do not want to evoke the biological aspect of it casually. So we have come up with a raft of phrases, from the Bibilical "with child" (which is very nice, I think) to the 20th-century expecting (which is the term I always heard in the 1960s. But today we like to think of ourselves as direct, casual, and nonchalant, if not flippant, when it comes to sexual matters; hence, preggers. But I prefer the more restrained "expecting", though I don't have any real problems with the word "pregnant" either.
And now my standard gratuitous Slovene interpolation: the Slovene word for "pregnant" is nose�?a, which is simply the feminine present participial adjective of the verb nositi, "to carry". So Slovenes say: "She's carrying."
rolig, that's an interesting theory. I never heard that—and would never have guessed that it's exclusively British. Hm. I have heard it used quite a bit by Americans—even by pregnant Americans—but I would guess it's one of those love-it-or-hate-it words, so hasn't entirely caught on with the general population.
Also, I wonder if it ties in with your (it was you, right?) theory about Americans not wanting to sound impolite even when giving orders. Pregnant is not the nicest of words, is sometimes even used as an epithet, so shortening it to something cutesy like this might appeal to some Americans' distaste for in-your-face-ness. (To coin a phrase.)
For the record, the reason I first listed this is because it was my original 'least favorite word.' I loathe this word. I don't have anything against pregnancy per se (though it would be much cooler if humans were ovoviviparous), I just really, really hate the way it sounds.