At my hospital (which is in sydney so not much help sorry) you were able to request a private room and they would "see what they could do"... You were more likely to get it if you have private cover, as that would pay for the room... or if you had twins or more, or if you were in hospital for an extended period of time and a few other reasons. They said they would not normally give the private rooms to a public patient because if a private patient showed up they would have to give your room to them.. but it could vary between hospitals. They also had a birth centre, where you got your own room and you stayed the night in the room you give birth in. Maybe that could be an option?