Lost_luggage34: One would like to think it has been updated somewhat these days as I am merely providing an historical description.
The BA website now takes a direct feed from the ops system, as far as I can tell. Mobile phones, of course, now provide a check on this.
The last time that I was inadvertently checking it, I was on the phone to my partner who was on a delayed flight waiting to push back at LHR. I was refreshing the BA website screen from time to time to see whether there was any updated information coming from that.
Suddenly, somewhat earlier than the BA website estimated time, my partner said, "I have to switch off now because we've just started moving" (with engine start to come next). I pressed refresh and the screen immediately displayed the current time as the "departed" time.
