Seems to me like there are two separate issues here:
1. Getting an appointment for Down's Screening in the public system: If you have registered at a public hospital and called/emailed them, they will contact you. I recall for my son I had the same panic but they did call me back and it was arranged. However, that was in 2010 and they weren't providing it free of charge to all women in the system. As of last year (2011), they are offering it free to all eligible women registered for antenatal care in the public system but I guess this means there has been a flood and thus the first-come-first-served. However, since your wife is over 35, she should get priority.
To sort this out, you could try going down to the obstetrics unit you registered at to talk to them again but at least in the case of Prince of Wales, the unit that does the scanning is a separate one so you'd do better to try to call/email/contact them directly.
Just in case, I'd try to make a booking privately. Honestly, Dr Yu is quite busy... not sure you'd get an appointment at short notice (the reason I didn't see him in the end; I needed to make my appointments at shorter notice) but you could explain the situation or see if they can recommend someone else. Being a hospital polyclinic they will have a panel.
Same with Baptist Hospital... you could call the hospital and ask who can fit you in for the one-off procedure.
2. Your wife throwing up and not being able to eat. For this, if you want to go public, I think you'd have to go to the A&E and yes, unless there are other complications, it will be a long wait. A friend with a similar problem waited something akin to 3 hours or more at the end of which they sent her home with some anti-nausea drugs. Another friend ended up being hospitalised for dehydration... not sure how long she had to wait though but obviously her case was considered more serious. So it depends how they assess the case, not necessary she will be whizzed through because she's pregnant. The public hospital I went to (Tseung Kwan O) had a notice up at the obstetrics clinic saying if you need to make an unscheduled appointment, you should go to a private doctor (!) or the A&E... they have too much demand. However, they did once advance my appointment when I called and explained that a private doctor had certified my baby as breech and a quick decision on delivery would have to be taken. So you can try the obstetrics clinic but I suspect in this case they will head you off to A&E.
Honestly, I'd just try to find a private doctor to assess your wife and if she's really in bad shape, a letter from that doctor might speed up the process in the public system.