Welcome to SG and this forum!
IVF at private hospitals normally will cost more but you get what you paid for. The waiting time for each appointments would generally be shorter and for most private hospitals, the scans will be done by the IVF doctors and you do not need to go to the pharmacy and queue up for medication. Basically, it's faster and more personalized.
IVF at public hospitals will cost lesser but waiting time tend to be longer. For SOME public hospitals, the scans are done by the sonographers and not the doctors so you may have to schedule different appointments for the sonographer and doctor eg. you see the sonographer at 8.30am and queue for scan and then later have to wait again for your doctor. For NUH, scans are done by doctors though. For public hospitals, you will have to queue at the pharmacy for certain medications and thus more waiting time.
I've done IVF at both private and public. I appreciated the time I saved at the private hospital but I do feel that the public hospital is just as good if you don't mind the waiting time. In fact, if you are a private patient at a public hospital (without subsidies, which will be in your case), then the waiting time may be shorter than a subsidised patient. Furthermore, public hospitals may have more resources.
These are some well known IVF drs.
Private Hospital
SF Loh - Thomson Fertility Centre
Dr Sheila Loh - Raffles Fertility Centre
Dr Ann Tan - Mount Elizabeth Orchard
Dr Roland Chieng - Virtus Fertility Centre
Public Hospital
Dr Tan HH - KKH
Dr Sadhana - KKH
Prof PC Wong - NUH
Dr Yu Su Ling - SGH
I did my first IVF with Dr Sheila Loh at Raffles and had a good experience.
I'm currently with Dr Shakinah Rauff at NUH and I absolutely love her too. She's so cheerful.
I hope this helps!