After reading / skimming most of the posts, I blame the Heathrow / UK practice of not displaying the gate until ~45 minutes before takeoff, even though 95%(?) of the time the gate is already known by 2-3 hours before and doesn't change. Plenty of airports are able to assign gates 24 hours in advance or more, I know LHR doesn't want to work like that, but if the OP had been given a gate on their BP at check-in (I see no indication they arrived at the airport ridiculously early), they would likely have headed to that gate at the appropriate time, so even the gate changed at the last minute, there would have been sufficient time to reach the new gate.

Someone mentioned above that they hadn't heard of this being a recurring issue, but I am sure I read about a similar occurrence here on FT in April or early - because on my last BA flight from LHR in early May, I remember knowing that I had to be wary of the gate information on the lounge board, and therefore left the lounge earlier than I wanted to because the lounge board was not showing gate info, in order to check the boards outside (and with a child + paraphernalia we weren't going to go back in) which did have the correct gate info (ok, it's possible that the gate info managed to appear in the 2-3 minutes it took us to get out of the lounge).