Perfectly possible, and I personally know that it does happen.
On my flight to Riyadh with BA last year, the skipper hailed from Florida (and very nice chap he was too-allowed me to jumpseat the landing at RUH).
On an Airtours 757 flight to Alicante from Birmingham, again, the Skipper was American, this time from Rochester (NY?)
I don't beleive you need to take up British citizenship-you merely need the right to live & work in the UK (work permit).
AFAIK, the UK regards Pilots as Professionals, and under this category, it is not difficult to obtain a UK work permit, all other factors remaining equal.
Having said this (and specifically regarding the US & UK), I would have thought the net movement of pilots would be westwards, towards the American airlines, which appear to offer much better money.
Having said that, the BA skipper on the flight to Saudi said he had wanted a change of lifestyle, and enjoyed living in the UK. BA perhaps offers attractive routes or a/c to pilots of other nationalities-I really don't know.
There are plenty of Americans working for Cathay and Emirates (or there used to be, anyway).
p.s. Now that I remember, the skipper on my AA 738 flight to SEA from ORD was British-funny hearing a Lancashire accent so far from home!