>>As far as I can tell Its American at the moment, followed by United and then British Airways. <<
Not quite, sir.
American, United, then Delta. British Airways figures further down the list, and ranks FIRST on being the Biggest International Airline, given that most BA flights (including those to Europe) are regarded as 'international'.
And the US/UA deal really is as dead as Kiwi Airlines.
FLY DELTA JETS and sail UNITED STATES LINES
N 8 6 3 D A