I hate to burst you little bubble there, Ianhol, but in fact there still are quite a few unemployed British pilots around (albeit with low hours). Also the airline sponsorships will be very hard to get in to, simply because so many people will apply that they can take the very best. But ofcourse you are the very best (sorry)..
As for the initial question, yes indeed there are many foreign pilots at certain airlines (Cathay Pacific, Emirates, Singapore, Royal Brunei etc), but thanks to the very stringent US immigration laws there will be absolutely no danger of many foreign pilots getting a permanent position in the US. Likewise it's not too likely that many US pilots will find a job in Europe with a European airline. This world is still a long way away from a genuine global job market where anybody could get a job anywhere.