NTU is considered as an autonomous university (gov.sg — Directory). They are not public or civil entities per se, but private. Thus employees under them are not public or civil servants.

I think it usually takes HR some time to process the approval for your application, length of which depends on the process and approval layers involved. You should probably receive feedback from them within two months. But I think you can give them a call to politely enquire on the status of your application, dont think there's ánything wrong with it. Good luck anyway!