Is there a word or phrase for a nursing mother not biologically related to the baby she breastfeeds?

They're called wet nurses. Wikipedia

• A wet nurse is a woman who breast feeds and cares for another's child. Wet nurses are employed when the mother is unable or chooses not to nurse the child herself.

• Wet-nursed children may be known as "milk-siblings", and in some cultures the families are linked by a special relationship of milk kinship. Mothers who nurse each other's babies are engaging in a reciprocal act known as cross-nursing or co-nursing.

  • For years it was a really good job for a woman. In 17th- and 18th-century Britain a woman would earn more money as a wet nurse than her husband could as a laborer. And if you were a royal wet nurse you would be honored for life.

That would be a nurse or a wet nurse:

nurse: 1 (a) a woman who suckles an infant not her own : wet nurse

wet nurse: a woman who cares for and suckles children not her own

Both definitions from Merriam-Webster Online.

Oxford Shorter (1933 edition)

Wet nurse, wet-nurse sb 1620 A woman who is hired to suckle and nurse another woman's child.

Wet-nurse verb transitive, to serve as a wet nurse.