Being a UK citizen you do not qualify for the Green card lottery. There's a list on the lottery website, which rules out citizens from specific countries. Uk is on the list.
I'm a UK and canadian citizen, but I'm married to an American.
SWA is my first choice in an ideal world, what with my wife already back in the US studying.
Just got to work hard now to obtain my command on B738, get the hours required along with the green card, and cross my fingers I get a chance to interview for a position I'm already in now-on the type I'm on now.