The umbilical cord is covered with a thick protective coating known as Wharton's Jelly. This is like gristle in texture and prevents the baby from compressing the arteries and vein that run through the cord. So the cord being wrapped does not usually pose a problem for baby.
At the time of birth, once the baby's head is out, the midwife or doctor will check around the baby's neck for the presence of the umbilical cord. Typically the cord is wrapped loosely enough for the cord to be slipped over the baby's head. If the cord is wrapped multiple times this may take awhile. Typically you will be asked not to push for a minute while this happens.
Sometimes the cord is wrapped too tightly and the cord will be cut before the baby can be born. This is done by your midwife or doctor by placing two cord clamps and cutting between them. This necessitates the baby's birth fairly rapidly since it is no longer getting nutrients from the mother via the placenta.
Occasionally the baby will be born so rapidly that neither of these methods can be employed. A skilled practitioner will hold the baby's head very even and close to the mother's body as the baby's body is born through the cord. This almost looks like the baby flips out as it is born.
The cord being around the neck usually requires no additional monitoring of the baby or mother.