Will my body know to start making milk in time for the baby to come?

This is my first baby and I am 37 weeks today, but have had no sign at all of leaking colostrum or anything like that. I want to be able to feed my baby right after he is born, but does it ever happen where you aren't even making milk to be able to feed? Or will my bbs automatically know when he is born that it is time to get busy and feed the little guy? Thanks!