Can you find a new flight to Nice (or another airport sufficiently close)? Doesn't have to be with BA. If possible come up with a couple of alternatives. Then ask BA to rebook you on that. If that doesn't work, book yourself on that (assuming you need to go) and attempt to claim compensation from BA after the event.

Looking at ITA Matrix there are a fair number of flights. Swiss via ZRH. KLM via AMS (don't attempt with checked luggage), and even a direct BA flight later in the day (BA356 at 17:45). All these have seats available according to matrix. There's also AF 7381 (Air France) at 17:35 which definitely has seats. At least some of these (particularly the BA one) BA should be able to rebook you onto.