Thank you for everyone's input and feedback. I do recognise that is my responsibility to be on time, rather than being in time. The gate was actually at A level, which I do believe I have a realistic probability of getting through with only one rucksack.

The gate agent did tell me the gate is closing in a few minutes, which I safely assumed was the boarding gate. Instead, he was referring to the security gate. I should not have waved goodbye to my friends and lingered at the terminal for a few minutes.

I booked a new flight through Avios today and I will see if there's anything that Customer Service would do in relation to my prior flight. It's extremely difficult to get hold of BA to try to rebook for another later flight since they said they removed all ticketing desks at Terminal 5.

Does anyone have similar experience and get their avios/taxes back?