The idea that the press won't know fairly instantly what hospital Kate is taken to to give birth is absurd. Someone, whether it be their own staff, palace staff, a police protection officer or a staff member at the hospital concerned will tip off the press, whether on Twitter, or using text messages or phone calls. It will be impossible to keep such information out of the hands of the media. The only reason the press weren't tipped off about the pregnancy (sorry, but I think despite what he says the photographer from Ikon just got lucky, even a stopped clock is right twice a day) is because William and Kate had told no-one, until hospital treatment was required.
The first tradition on big occasions like this, whether it concerns births or deaths in the RF, is for a physical statement to be hung on the gate of BP (
here is the announcement of Prince William's birth). Of course, in this day and age a statement will also be issued to the press using up to date means also.