Perhaps you might want to call your preferred centre or anywhere nearby your area, let them know of the urgency to have your LO in the waitlist and call you soon as they have slot. Let them know you are taking care of LO, you are working and caring for her alone therefore if she has no slot with any PN/PG then you wont be able to go work.
Also another tip is to go individual preferred school and apply and input your earliest date as possible. Do not go ECDA (I tried few times, taking so long) hence I went individual school website apply. And I was lucky to have a slot in August for my second born. I told them I need it ASAP or I wont be able to start work.
Just FYI, doesn't mean housing estate with BTO do not have vacancy but each centre has a ratio of teachers to student hence why no slot. Also some centre that I know of are having a lot of vacancy as they just open and sometimes in a month, you will have more than 2 kids enrolling. So it depends on luck though.
itsallabout_mytwokids