You might consider keeping BA and "see what happens with regards flight bookings". From Athens, one option if travelling J is to fly ATH-DOH-JFK on QR. DOH might be the closest hub airport. (Actually it probably RJ at AMM but I am not too much of a fan but it can be rather cheap). I am unclear how the QR or RJ FF scheme works but if you get back to the UK now and then, you should still make the 4 qualifying flights a year.

To be honest, you might wish consider TK for flights to the US as well on *A. Decent service and food and IST lounge.

QR/TK both fly to GRU direct from DOH/IST so you can by-pass the US if you wish. The DOH-GRU flight, continues on to EZE.