Hi Noella,
Congratulations. I had my first child in the UK and my second here in Hong Kong. They were completely different experiences.
I went down the private route as luckily everything was covered by insurance. The first thing I did was get a doctor. I can recommend Dr Dawkins. He's very gentle and will answer your myriad of questions. With him I bought a package, so that the antennatal visits were covered. I know he has now moved practices, so not sure whether he's still doing packages. I chose to give birth to my daughter at The Matilda. Having got the doctor sorted out, I then contacted the insurance company and they contacted the hospital directly to set up an account so that side of things was paid for. My husband still had to fill in a load of forms when we arrived in labour, but he didn't have to pay anything.
My doctor went on a course during the week of my due date and Dr Sally Ferguson actually delivered her. She was absolutely fantastic and I can't praise her highly enough. We are intending to have a third and I shall use Sally again.
The other excellent part of my labour was Hulda from Annerley Midwives. She was our doula/midwife. She came to the house when I went into labour and accompanied us to the hospital. She stayed for the entire birth and was ABSOLUTELY BRILLIANT. There were a few reasons that I chose to have a doula; I was new to Hong Kong and needed someone who knew the medical system very well and also I had heard that C-section rates in HK were alot higher than in the UK, so I wanted someone who was medically trained to fight my corner if necessary as I was determined that I didn't want a C-section. The package that I paid for with Hulda also meant that she visited my house after my daughter was born to weigh and measure her; they don't have health visitors like we do at home.
I should say something about the hospital. As I say my son was born on the NHS in Kettering General. My daughter was born in a 5 star hotel that they call the Matilda. The hospital was wonderful. They made every effort to follow my birth plan and I got the birth that I was looking for. The care and attention that I received from the midwives and Dr Ferguson, I'm sure made my recovery alot faster than with my first child. And the food is fabualous; it's actually prepared by a chef who used to work for one of the big hotels in Hong Kong. So from my experience, if your insurance will pay for it, I would'nt hesitate to recommend the Matilda.
I hope that helps a little. If there are any specific questions you have just ask. I came to Hong Kong, 6 months pregnant and managed to arrange everything in time. Just make sure you ask lots if questions and you'll get the answers you're looking for.

Jools