What is the ruling of Islamic law regarding a father taking custody of his children (two daughters and a son) when they are living with their mother, who is a nurse?

The mother has a greater right to the custody of her child before the age of discernment, unless she marries or has an impediment.

They cited the Hadith of Abdullah ibn Amr, in which a woman said: "O Messenger of Allah, this son of mine was a vessel for my womb, a drink for my breast, and a sanctuary for my lap. His father divorced me and wants to take him from me." The Messenger of Allah, peace and blessings be upon him, said: "You have a greater right to him as long as you do not marry." Narrated by Abu Dawud and graded as sound by Al-Albani.

As for after the age of discernment, the preferred view is that if a boy reaches seven years of age, he is given a choice between his parents. As for a girl, after the age of seven, she stays with her father. In cases of dispute, the matter should be referred to the Sharia courts for adjudication.