Lots of good information here, however, although there may be no restrictions on Pilot jobs for those with permanent residency (I assume you mean a Green Card)
It is not easy to get a job with a US Major without being an American Citizen.
Not saying it's impossible but given the choice between you and a US citizen they will nearly always choose the latter and claim security considerations post 9 -11.
My point is, if you decide to move to the US and stay here, you would be a lot better off becoming a US Citizen, not just for career purposes.
You would not lose your British Citizenship, or right to a UK Passport if you did this. Although the US would not recognize any other citizenship you can never 'lose' your original one.