Did this mom have artificial nails prior to delivery? If so the top layers
will keep peeling off until the entire nail is replaced from the base up (3
months or mo).
Nail salons have many products to treat peeling. I've found a couple drops of
cuticle oil (Canuba or calendula liquid form?) rubbed  into the nails before
bed helps. Lansinoh might even work.
Polishes and nail strengtheners even hand lotion will only dry it out more.