If you call they will only tell you the stuff they have to tell you. Here, it has a lot more to do with who you know rather than what you know.
Here's a shot in the dark but may interest the BA guys. What about CX? You have the right to live in the US and the UK so you could go to either at one point. You could be based in the US when you want and elsewhere when you want. The pay is somewhere between US carriers and European carriers. The culture is also probably what you are used to as well. It seems to me that with your time and experience they would probably invite you right in, especially with the 777 time.
Just a thought but it might work out for you.