Pretty much all the private hospitals (on HK island at least) are full at this point for an August due date. The best you can do is, as noted by others, find a doc who is resident at one or more of the private hospitals and hope that a spot opens up. Some of the hospitals (Matilda is one) have waiting lists, while others don't. Adventist is an excellent facility and doesn't have overbooking issues any more than any of the other private hospitals - they're all booked up months and months in advance and only very rarely do you hear about someone not being able to have their baby at a private hospital AFTER they have been confirmed for a spot and paid the deposit because they are actually over booked.
Lots of posts on here cover the public option and Queen Mary specifically (take a look at the 'Similar Threads' links below). Most people are happy or at least okay with it, and the quality of care is generally quite high, but not nearly as much privacy (or as much visiting time) as you would have in a private hospital.