I used to work at London ATC and would bet my pension that BA does not get priority. As far as ATC is concerned you all pay the same charges so all get the same service.
Heathrow is so busy all day that a straight-in would only be offered very early in the morning or late in the evening. I am not sure what the circumstances were but maybe you had to hold because you arrived before 0600 and did not have an airport runway slot until after that time. The BA aircraft behind you was probably able to land before 0600.
As far as departures are concerned, exactly when you are cleared for take-off depends on the other aircraft their vortex wake category and which SID they are following. There has to be at least 2 minutes between aircraft following the same or similar SIDs e.g WOBUN/BPK, CPT/SAM.