So ultimately as you are more than aware it does come down to the passenger.

That said there is no recorded gate closed time, so it may be the case IDAHO wasn’t being updated by the gate lead. The official timings show the gate at A10 opened at 1059. Being a coaching departure to stand 591 the doors closed on the aircraft at 1233. There is a note that a re crewing was required and the pilots were running late on the delayed BA1305 which arrived on stand 501 at 1137. There are 52 minutes of delay allocated to this. So it took under an hour to get the pilots out over to stand 591, load up the coaches, travel to 591 and board the aircraft.
The crew recycled the brakes to indicate they were fully ready and just waiting on ATC at 1237 which is the departed time you see on the app. Actual clearance to start was granted at 1253 vs a target start up time of 1255.
Boarding of coaches most likely commenced around the time of the inbound flight crew or shortly thereafter, but as you are already more than aware there was plenty of time to meander down to A10. Sorry.

I am however surprised no lounge announcement was made given they’d know you were there.