I'm thinking the Palace have told a fib about the due date and possibly location of the birth so Kate can get through it all in peace. It's what I would do and it would be hilarious to have all those paps outside the hospital for days for nothing.
Doubt it. They know the worldwide appetite for this kind of thing. It would be a huge anti-climax if they said "sorry, she already gave birth". She'll have the baby at the hospital, they'll release an official photo of Kate, Will and the baby, and then the paps will get their money shot when they leave the hospital and get into the car, although hopefully they'll be considerate enough to switch the flashes off.