Is it permissible to convince the father to return the children to their mother, what is the ruling on the action of the maternal uncle who exposed his brother's wife's infidelity, and what is the legitimate ruling regarding who should have custody of the children after this incident?

What your uncle did by recording and filming his brother's wife is an act of disobedience and a sin, and he must repent, because he exposed her. A Muslim is commanded to conceal the faults of his brother. He should have advised her, instilled the fear of Allah in her, and threatened to inform her husband. As for custody, the mother has a greater right to it as long as she does not marry a stranger. Her custody is not forfeited except by her open transgression. In this case, the mother has the primary right to custody.