amazing that BA can get away with it really given that in this day and age you are not allowed to be sexist, racist etc but heightist is just fine.
For what it is worth I am 6'5" and the only issue I have ever had was whilst doing my initial training at Oxford back in the mid 90's. There were three of us on the tall side and it was decided that we were too large for the T67's. The result being that the school got rid of them and bought a couple of brand new Zlin 242's.
The much greater size issue that I could see was a short round airline student who needed booster cushions to get high enough and close enough to the controls. His gut then almost stopped him from getting the yolk fully aft.