I agree with
guv1976 I would fly BA LHR-BNA, and then consider flying AA/BA from MEM-XXX-LHR. I'm relatively close to MEM and prefer connecting at DFW for the better lounges at DFW (vs AUS, CLT, or PHL) and the longer flight time over most other options (the short overnight transatlantics don't leave a lot of time to sleep) but that's just personal preference on connecting airport.