Definitely BA. Food is better than AA, lounges are much better, airport experience is nicer. If you like, you could even split the difference and book yourself on AA LHR-ORD with a long connection after arriving from LIS so you can use LHR T3 and enjoy the CX, QF and BA lounges.
In the other direction you should fly on BA metal ORD-LHR so you can benefit from EC261 compensation in case of delay/cancellation in both directions.
Not sure why people complain about CW on the A380 — just took it from LHR-BOS and loved it.