There is no fast track, the trick is not to join the queue if it's long! Which means about a dozen people. The security (and passport check) is at the BA gate and unless you happen to see the best gate entrance open (e.g for the TK flight) and empty then it is what it is. Typically 5 to 10 minutes. However they do advise you in the lounge when it is time to leave for the gate and while there is no need to rush at that point, so long as you leave within a few minutes or so then it tends to work out fine. Not worth overthinking.