what kind of HK id cards do you have? does your wife have a valid visa to live in hk?
if she has a valid visa to live in hk (FDH, work, dependent) or is a permanent resident, then she is considered an eligible person and can make use of the public system for next to nothing in cost.
if she does not have a valid visa or is not a permanent resident, she is INeligible and cannot get the discounted costing. she would still be able to deliver, but it would cost you a lot more. ($3900/night vs $100/night, if memory serves me correctly).
your visa status has nothing to do with it. you are not the patient, your wife is.
if she is eligible to use the public system, which hospital you use will depend on where you live.
yes, you will have enough time. the public hospital, so long as it has maternity facilities, will not turn your wife away if she is in the middle of labour.
you will need to think about your wife's visa status, your visa status and the status of your child. merely being born in hk does NOT qualify your baby to be a permanent resident or a citizen. if either of you are permanent residents, then there is a possibility your child could get permanent residence here. otherwise, your baby will need a visa, just like you and your wife.