How should I start the birth control patch after having a baby?

Wait until 6 weeks. Assuming you have not had intercourse yet and are not breastfeeding, start the patch the first sunday after you are 6 weeks postpartum.
3-6 weeks. Ask your doctor even before your delivery. Depending on your health and risk factors, you can restart the patch in 3-6 weeks after delivery. I hope that you are planning to breast feed--it's the healthiest thing for your baby and easiest for you. That will also determine when you can restart, and which method is safest.