Soft product wise, CX wins over BA by a mile. Better IFE, better food, warmer service.

Hard product wise, I'm completely fine with CX, but large people have problems with their herringbone seats which are not wide and are a bit confining.

Also, you will not have to change terminals in LHR.

Personally, it's a no-brainer for CX. You will be sorely missing CX J on your connecting AA flight, even if you're in F.