There certainly are plenty of Brits and Europeans working in the USA, and they'll have got there by one of three ways:
1) Getting their Green Card sponsored by an employer because their skills are in short supply and staffed cannot be sourced in the USA - doesn't happen for pilots.
2) Getting their Green Card granted by the US Government from the annual quota for oppressed countries or those with special ties to the USA (eg Poland, Italy, Ireland) - doesn't apply to Brits.
3) Having acquired American citizenship by birth or through marriage to an American (in which case they would have to renounce their previous citizenship), or by being granted 'resident alien' status through marriage.
So there you are, all you gotta do is marry an American and the chance to work for UA/AA will be yours. Yanks, you're more than welcome to come over here and try for a job, you just gotta marry a citizen of the EU, acquire European citizenship and convert your licence to a JAA one. Couldn't be simpler!