Originally Posted by
corporate-wage-slave
BA can actually decline to rebook you
They'd probably be in breach of contract if they did this, and there might be remedies outside EC261 (Consumer Rights Act perhaps, but OP is travelling for work).
People understandably don't have the confidence to walk to UA, book a cash fare for replacement travel to e.g. IAD, and recoup the costs later. It would be nice if someone did this and got a binding judgment in his or her favour.