The OP has a valid question, although I've never experienced that situation myself. So, assuming the last day of the card validity follows the time/date zone, there might be a small step to circumvent the awkward situation. There are no BA flights in/out of Europe after midnight, IIRC. But in Asia, I have had flights from SIN starting after 00.00. Should the unusual situation occur, that ones card expires at 23.59.59, there could pose no problem to enter the lounge in good time before the bell chimes 12.

Be sure to put the soon-to-expire card in a Kevlar protective case, before the warning message turns on: This card will self-destroy in 15 seconds