congratulations on your pregnancy! I went through the same route with all four pregnancies; but i had a choice of gynae in mind from the start so I called in early (at about 5 weeks) to get the earliest slot — usually the popular ones has no slot until nearer to EDD and they might not be able to/want to see you if it's too late. for my last two I got to see my gynae only from week 20 onwards. cost wise is ok for private route, only issue is IF you have complications or need urgent o&g visit that will be super costly (mine hit up to $300 for one visit while I was still under subsidised route, so I can't imagine how much it will be for private). maybe do some research on a gynae you'd like and call in to schedule an appt first. good luck mommy!
Switched only at week 36 (called to book on week 32 but only got a slot for week 36) from subsidized to private. Did this to save some cost for all the prenatal checkups + urgent o&g visits as my pregnancy is uncomplicated. Estimated saving is about 2-3k. Something to note is that by week 32 alot of the popular Gynae are unavailble so you will need to see if you want to save some cost or do you want to secure a Gynae of your choice. Took a leap of faith and went with a Gynae that I randomly chance upon a review. But he turns out super good too!
I called at week 25 and got an appointment for a recommended doctor at week 32. You can consider to contact them after your gestational diabetes test!