I have done mixed classes with open jaw even - no problem
LHR-JNB in J and MAU-LHR in F
and this one dates really dates it
LHR-JFK in Concorde and BOS-LHR in F
LHR is BA's home base. BA's facilities will be best at LHR.
If you fly LHR-NRT in F, you get to use the Concorde Room (reserved for F pax only. Even BA Golds have to go to the First Class lounge).
J or F, you can try the to get an appointment at the Molton Brown Spa although it sounds like to need to get there about 5 hour before your flight to get an appointment for previous postings here.