CX F is vastly better than either BA or AA.

The BA/AA F decision depends in part on when the trip will be taken, and whether the new AA 777-300s are running yet. The AA F Suite provides greater space/privacy than BA's F seat, but the older 777-200 Suites are being held together with duct tape. [That said, some of BA's cabins are really filthy dirty.]

I agree the suggestion about travelling westbound --- the eastbound JFK - LHR overnight flight is really too short for a proper sleep (often just 5 or so hours in flight time).