The routine is generally to whisk the baby away for weighing, drying, and doing newborn checks while the doctors close the incision and the mother lies on the operating table. As long as the newborn is in stable condition, it can be brought to the mother and held skin to skin just as many would after a vaginal birth.
There are other things that can be done differently to make a cesarean section birth more meaningful. A friend of mine had a positive experience with her cesarean birth. Her midwife gave some really good suggestions to them before going in for surgery. One of those suggestions was for the father to hold the baby and not let the nurses have him back. When the nurses wanted to put the baby in the bassinet to take him to the nursery, the father gently refused and told them he would hold his son. Even after they told him he wasn't allowed to do that, he held tight to his newborn and insisted on carrying him to the nursery himself. He stayed with the baby when the mother couldn't be there.
There are options other than the normal routine, but you must ask for them. You need to know your rights and how to ensure you get them. Do you have any suggestions for expectant parents on what they can do in the event of a cesarean section?