Congratulations on your pregnancy and soon to be delivery!
I got here when I was around 5 months and I can understand how much more reassuring it will be get your doctors and hospitals lined up esp with a relocation also to manage.
Personally, I've had an excellent birthing experience here in Singapore. I delivered in Mt. Elizabeth, and I absolutely loved the facilities and the staff there. The doctors I got since I landed are the same ones I'm still with to-date. Here are their names:
1) Healthway Novena (Novena Square - 3rd Level) - I love their in-house GP. She's absolutely lovely, patient, and smart. I think she's Dr. Christine but may be wrong as I hadnt gone back for a year already. Price is very manageable and because the mall isnt very populated, you should be having short waiting times. She was the one who recommended both Dr. Yeo and Dr. Ee (see below).
2) Dr. Motoko Yeo (Paragon Medical Center - Motoko Clinic) - She's an OB-GYN. Its normal here for an OB-GYN to do the delivery for you then to have your pediatrician receive the baby and do the checkups in the same delivery room. She was absolutely patient and sane despite my begging her to "yank the baby" during my weakest moments.
3) Dr. Kenny Ee (Paragon Medical Center - Kinderclinic) - He's just wonderful. I had a pre-meet with him before my delivery and liked him right away. The staff in the hospital took care of calling him up when I was ready to deliver, and he assisted during my delivery then he made the first checks on my son as soon as my son was born. He has received evening calls and long distance calls from me, but he always responds kindly, promptly and efficiently. He is a bit pricey, but I trust him with my son's life esp as he's helping my son go through some medical conditions. He used to head the emergency dept in KK Women's and Children Hospital.