I personally don't see why it can't be done, apart from being inconvenient for the staff. After all, plenty of people are booked in for elective c-sections that go into labour and have to rush off to hospy to have their caesar.

Good on you for thinking ahead, and asking the question. I think it's perfectly reasonable thing to request. But don't forget that just because a body has gone into labour doesn't mean the bub is ready, not always the case (though it usually is)