Thank you all for your input. Nothing ever comes easily in this profession, does it?
From your replies, it sounds as though my best option is to return to my previous employer here in the UK, who offered me a command just before I left to BA. Spend a year or two back there accumulating PIC hours on 737s and then trying to make the move to the US.
A big step, I know, as it's very big decision to leave a carrier such as BA, flying longhaul, in order to take a shot at moving to the US a few years down the line.