I can't help shoving the kitten in amongst the pigeons.
I've flown to and from the UK (between both Gatwick, Heathrow and DFW) via American Airlines three times since getting my 2 year green card using a ticket in my married name and a passport in my maiden name.
I changed the name on my AAdvantage mileage account just after getting married. When I booked my first trip back to the UK a few months later I called AA and asked which name to book in, bearing in mind my AA account and green card are in my married name but my passport was (and still is) in my maiden name. The rep said using either name was OK, but it would be better to use the name that matched my AA account and also the green card. As long as I had the original marriage certificate with me on the trip it would be fine. So (admittedly with some trepidation) I decided to book in my married name - I wanted to be sure I would be able to get back into the US without difficulty since my ticket would match my green card.
So - as you know - when checking in these days there's a pre-check in security check desk in front of each airline's own check in desks ... At the pre-check in desk at DFW, the security staff simply compared the names on my marriage certificate with those on the ticket, passport and green card to confirm I was (and still am, I hope) the same person, added a note on the computer system to that effect and on I went. (In fact I have been a useful training exercise at DFW and Gatwick because a couple times newbie checkers have been shown how to handle my situation!) The actual airline check in processing has been quick because the information is all there in the system already.
I'm pleased to say that check in has been this easy every time, both in the US and the UK.
The last time I did this was in late January this year -
after the US passport rule changes came in and just over two years after we got married. Passing through US immigration, all they ever wanted to see each time was my green card.
In case you're wondering why I'm still doing this two years later ... My old UK passport ran out just a few weeks before I moved to the US, so of course I had to buy a new one and it's valid for 5 years. If it was free, or cheap to change a name, that would be different, but a simple name change costs just the same as a full renewal, and I'm stingy and just not prepared to fork out yet more money unnecessarily, so I'll keep on doing what I've done for another couple of years or so, unless either the USA or UK change their travel laws again in the meantime.
I guess I should make the following disclaimer ...
The above is a description of my experiences when flying with American Airlines to/from either Gatwick/Heathrow - DFW only. You should probably call the relevant airline yourself and then make a decision for yourself!
Happy journeys!