Hola Plaindriver -
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You will get a "Green Card" as you will be sponsored by your wife, so there will be no restrictions for employment by US air carriers for you. But read again what JW411 says here above... he is right about everything.
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I recommend you to continue your employment on the 737 in the UK, even if that requires you to commute from JFK, once or twice a month. I worked for 22 years (1969-1991) for PanAm, based JFK (living in Los Angeles) and the commuting was no better than commuting would be from UK.
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The US economy is in shambles, and the US airlines' low salaries (paid in devaluated dollars) offer very little in benefits, and retirement plans. There will be more airline bankruptcies and merges... After PanAm busted, I went overseas, and never considered going back, whether for work, or residence there. If you locate yourselves in the NYC area, I recommend you find a place to live in Connecticut, i.e. Hartford or Danbury...
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Should you elect to ignore my recommendations, the only airline that might have a future would be Southwest Airlines, and as they are 737 operator, they might hire you thanks to your 737 background.
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For you, the
easiest, no nonsense way to get a FAA ATP certificate (with your 737 background) is to simply pass the written ATP exam (2 to 3 days preparation and test), and get a quick all-simulator 737 type rating at the same time, to get a full FAA ATP combined with 737 rating. All inclusive, the cost would be no more than some $3,000 to 5,000, and would take you about 1 week to 10 days of your time, at worst.
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Happy contrails