There are two main windows for NUS and NTU undergraduate applications. If you hold international qualifications, the portals open around mid-October, the Q4 window that starts the next academic year's cycle. If you are a local A-Level student, you apply after your results: A-Level results land in late February, both portals open the same day, and the deadline lands around mid-March, roughly three weeks to complete everything. Poly and IB students have their own windows, so check the official admission guides early: NUS Office of Admissions and the NTU admission guide.
Whichever window is yours, the essays should never be written inside it. Start drafting in September or October.