BA certainly "screwed up royally" and should just pay out without further ado. But if they want to be difficult, they may be able to justify wriggling out of the regulation quoted by ExXB:
2. Paragraph 1 shall apply on the condition that passengers:
(a) have a confirmed reservation on the flight concerned and, except in the case of cancellation referred to in Article 5, present themselves for check-in,
BA could argue that the passenger did not in fact have a confirmed reservation on the flight, so could not be "denied boarding" within the meaning of the regulation. Now the reason the passenger did not have a confirmed reservation is that some fool at BA had cancelled it by mistake. But that doesn't change the fact that at the time boarding was denied, a reservation did not exist.