I love Babbage!
First of all, the difference and analytical engines have clear military applications (indeed, Babbage's initial funding came from the Royal Navy), so it's possible that Babbage won't want to or won't be allowed to develop it somewhere that's not the UK. But let's set that aside, because maybe Babbage does get angry enough with the RN that he decides to quit London for the colonies and casts his fate in with the United States. Of course, two full difference engines created by cribbing Babbage's notes were built and sold to the UK and US in 1860; both used them for compiling logarithmic and trigonometric tables. So maybe this isn't a big issue.
I guess the big question is who's funding it and why. OTL, Babbage's first difference engine had about 17,000 pounds (1/3 the cost of a heavy frigate) sunk into it before it was cancelled; that said, a Swedish engineer later built 2 working copies based on Babbage's work with minor adjustments, and sold them in 1860 to the UK and US, at about 1,000 pounds each; I don't know if that's because of advances in technology along the way, or the lower cost of Swedish labor, or what (maybe Babbage wasn't a great engineer when it came to building things physically?).
An analytical engine will probably be more expensive. But, apparently the purchase of the difference engine was due to a small group of Americans (mostly astronomers, but also a Coast Guard officer) convinced the American Association for the Advancement of Science that the idea was a sound one, and they actually corresponded with Babbage and sent someone to Europe to meet with him. Maybe they're sufficiently impressed by the difference engine they bought to start fronting money for the analytical engine. But the outbreak of the Civil War will probably see the project delayed until afterwards. It might be possible to push the entire enterprise forward in time as early as, say, 1855 or a little earlier, in which case we could maybe get one before the war.
But what does an analytical engine get us that a difference engine doesn't?
The programs that Babbage himself developed were for things like solving polynomials of fixed orders, factoring matrices, and calculating Bernoulli numbers and factorials (both are most easily expressed recurvisely). It could also have been used to do things like calculate pi and e, run simplish Monte Carlo simulations, factor numbers...
I mean, technically, it was Turing complete, so it could do anything a modern computer could, but it practice it was invented for calculating mathematical terms, and that's what people were thinking about when they thought to program it. Early applications will probably involve things like compiling tables, calculating trajectories of astronomical objects, trying to calculate more accurate long-range artillery fire by taking into account things like the curvature of the Earth, and other fairly abstract matters of math and physics. If it gets to atomic physics, then you could see cool stuff like trying to calculate shockwaves and quantum mechanical stuff.
But I don't think we're going to be seeing any William Gibson stuff arising from it, sadly. Mechanical computers just aren't fast enough for cool stuff like a panopticon.