Notified by what method exactly?
BA67 today closed boarding at 13:50 in theory, so 10 minutes after 13:40, but the captain requested pushback at 13:51, and is now taxiing for immediate take off. So I think the agent was a bit slow in closing up, but otherwise this is an on time departure. Edit: She is now cleared for take off, so if anything a few minutes early on target take off time
LHR and LCY flights can and often do leave up to 10 minutes early. 15 minutes at LGW. That's just a function of everyone being on board and no issues with cargo etc.