Hey,
I just gave birth in November so I've gone through all this registration stuff. Let me just share with you my experience and other info that I know.
1) First of all, as what the other comment already mentioned, hospitals here won't schedule you for check-ups until you're 11 weeks pregnant. In my case, I found out really late that I was pregnant so I didn't go for check up until my 5th month already, so NO, it's never too late.
2) If you want to have your delivery at Queen Mary, your check-up will be at Tsan Yuk Hospital. You need to call them to make an appointment and you'll need to bring the following: proof of address, HKID or passport, referral letter from private doctor. Once you've registered, all other check-ups will be a breeze. The clinic will give you all the information that you need.
3) Delivery at Queen Mary: no, there's no shortage in beds. But again, all public hospitals here will require you to have done ante-natal check-ups with them if you want to have your delivery there. (Tsan Yuk Hospital is like a subsidiary of Queen Mary)
I would suggest still going for check up with a private doctor if you want to see an ultrasound every check up. Gov't hospitals only check your baby bump's growth and the heartbeat of the baby. I know a good private OB which is not too expensive, HKD 800 per check-up. Send me a PM and I can give you his info.
Also, if you're still undecided if you want to deliver to private hospital or govt hospital, let me tell you why I opted for govt hospital:
1) It is A LOT cheaper. Private hospitals charge from 25-30k HKD for childbirth package. In addition to this, you'll have to pay for any meds, anesthesia they give you and the daily rate for your room which is like 2-3k HKD.
2) Based from my friend's experience and what my nurse (from a private hospital) told me, should there be any complications during delivery with you or the baby, they will most likely end up sending you or the baby to a govt hospital anyway. There are some meds that are only available at govt hospitals.
That pretty much got me 100% decided to go to govt hospital.
Sorry for the long response. Hope this helped!