Generally most places don't let partners stay, or only stay one night. This is largely due to keeping the maternity centre friendly to other mums, who might feel rather uncomfortable coming across a random guy walking the corridors in the middle of the night.
I had a c-section at North Shore hospital, and transferred up to Warkworth birthing centre after 24 hrs as soon as I was allowed - it really was a completely different world with a quiet setup and very helpful nurses/midwives on hand. They even took my son for 3 hrs on the 3rd night to allow me to get some sleep! (he had bad reflux, and even they had a lot of trouble getting him settled to sleep!). With all birthing centres/maternity centres women who are able to give birth their have priority over anyone transferring, so simply depends on how many women give birth at the same time as you. Have to cross your fingers and hope for the best!