When does your milk dry up ?

I was wondering when your milk dries up. I want to get my nipples pierced I'm currently 37w so after I have my daughter I wanted to get them done but I know you milk supply comes in a few days after birth. I don't plan to breastfeed so when does it dry up so I can get my nips pierced? should I wait a week or two after I have her? or should I wait longer