Well, if Wikipedia is to be trusted, The Ministers of the Crown Act 1937 gives the Speaker the power to decide which is the numerically strongest party not in government, and who the leader of that party is, should either of those things be in doubt. However, he doesn't have to use this power: there was no Opposition during the WWII National Government.
The question becomes significant should the PLP pass a motion of no confidence in Corbyn. At that point there's a potential constitutional crisis within the Labour Party: the conventional thing would be for him to resign, triggering a fresh leadership election, and the deputy leader would be acting Leader of the Opposition until the election was complete. If Corbyn didn't resign after losing the confidence of his party, there's no telling what would happen, but a split seems possible. However, that might well leave Corbyn in possession of the party organisation, which would put the anti-Corbynites in a curious position.
no subject
The question becomes significant should the PLP pass a motion of no confidence in Corbyn. At that point there's a potential constitutional crisis within the Labour Party: the conventional thing would be for him to resign, triggering a fresh leadership election, and the deputy leader would be acting Leader of the Opposition until the election was complete. If Corbyn didn't resign after losing the confidence of his party, there's no telling what would happen, but a split seems possible. However, that might well leave Corbyn in possession of the party organisation, which would put the anti-Corbynites in a curious position.