AAdvantage allows flying BA from Canada, Mexico or Carribean trans-Atlantic, but not from US. This may make the trip less convenient.

I fly paid BA F and AA F regularly, and I have chosen to fly AA on a paid F ticket when I could have flown BA; obviously I don't think BA F is a "million times better" than AA F! It is better, but only slightly (maybe 10% overall) and there are many things that AA does better than BA (Bose noise cancelling headsets, for example). BA F used to be quite a bit better but has gotten a lot worse in the past year, particularly on the shorter routes like US east coast to LHR. I suppose for a "once in a lifetime" experience I would pick BA over AA, but on a different route (LHR-HKG, LHR-JNB or LHR-SYD, all seem to have better F service than the US routes).

As for your original question of daytime flight vs. overnight, in terms of enjoying the flight the daytime is far better (twice as good at least). But you lose a day (you arrive in London in the evening and have to pay for a hotel, while on the night flight you sleep on the plane). Of course the positive aspect is a better night's sleep on the ground. Usually my schedule prevents taking the day flight, but given the choice I would pick that. Very nice from either JFK-LHR or ORD-LHR (similar service though ORD is a bit longer and has the FL while JFK does not).