This is referred to as "delivery en caul". Complete expulsion of the baby signals the successful completion of the second stage of labour. Some babies, especially preterm infants, are born covered with a waxy or cheese-like white substance called vernix. It is thought to have some protective roles during fetal development and for a few hours after birth. The second stage varies from one woman to another. In first labours, birth is usually completed within three hours whereas in subsequent
labours, birth is usually completed within two hours. Second-stage labours longer than three hours are associated with declining rates of spontaneous vaginal delivery and increasing rates of infection, perineal tears, and obstetric haemorrhage, as well as the need for intensive care of the neonate. Third stage: placenta delivery
The period from just after the fetus is expelled until just after the placenta is expelled is called the third stage of labour or the involution stage. Placental expulsion begins as a physiological separation from the wall of the uterus. The average time from delivery of the baby until complete expulsion of the placenta is estimated to be 10–12 minutes dependent on whether active or expectant management is employed. In as many as 3% of all vaginal deliveries, the duration of the third stage is longer than 30 minutes and raises concern for retained placenta.Placental expulsion can be managed actively or it can be managed expectantly, allowing the placenta to be expelled without medical assistance.
It is advised that the newborn be placed skin-to-skin with the mother following vaginal birth, or as soon as the mother is alert and responsive after a Caesarean section, postponing any routine procedures for at least one to two hours. The babys father or other support person may also choose to hold the baby SSC until the mother recovers from the anesthetic.The WHO suggests that any initial observations of the infant can be done while the infant remains close to the mother, saying that even a brief separation before the baby has had its first feed can disturb the bonding process. They further advise frequent skin-to-skin contact as much as possible during the first days after delivery, especially if it was interrupted for some reason after the delivery.La Leche League advises a women to have a delivery team which includes a support person who will advocate to assure that:

The mother and her baby are not separated unnecessarily
The baby will receive only her milk
The baby will receive no supplementation without a medical reason
All testing, bathing or other procedures are done in the parents roomIt has long been known that a mothers level of the hormone oxytocin, sometimes called  "the love hormone", elevates in a mother when she interacts with her infant. In 2019, a large review of the effects of oxytocin found that the oxytocin level in fathers  that engage in SSC is increased as well.