Oh, I remember one time waiting with running engine for our gate at T4 to be cleared. It was occupied by a BA 744 which was obviously ready to be towed but according to GMC BA had no tug. While waiting though (for 45 minutes !), we could see several other BA aircraft towed away from their gates just in time for another Speedbird (which all had just touched down) to dock.
Clearly not well versed tn the BA way. I bet the guys on the 74 were just as frustrated as you. There is no preference for anyone at LHR!