For my first child, delivered by cesarean section, the baby was cleaned up a bit, examined briefly by a doctor, and then given to my husband (I was still on the surgical table, of course). My husband held the baby up by my face so I could see her.
For my second child, delivered naturally, the baby was cleaned up a bit and examined. He was having some trouble breathing, so they allowed me to hold him for a few minutes, but then needed to take him away to get his breathing straightened out (my husband went with the baby). Fortunately, it was nothing major, and he was fine in a few hours.
For both children, they were not fully bathed/cleaned for about a day (24 hours) after birth.