PE in the US as an international after B school

Yes you can apply, make sure to attend a STEM-designated program so your visa is for 3 years and not just 1 year. The visa situation beyond that 3-year period is very difficult (H1B odds are ~10%), so many firms do not hire internationals at all because there's a high chance they'd lose you in a few years. You can still land, but be prepared to work much harder for it than your American classmates and don't be offended if you aren't getting interviews with a strong resume. It's the international visa.

PE runs the gamut from top MFs (probably out of reach unless your prior experience was MF) to small 10-man PE shops (could probably get there without b-school). Hard to give a precise answer without knowing what size firm you were previously at/whether it was in London or a smaller market, and where you are going to B-school. 

Not sure how set you are on B-school, but a more permanent way to transfer over is via the L-1 visa which is not a lottery and essentially just an office transfer vissa. Get a job in London working for an American company, and in 2-3 years ask to transfer to the NY office. Much easier, no $$$ of B-school and you aren't on a 3-year bomb facing impossible H1B odds like you would be with your B-school visa.