What should a mother do regarding her wealthy ex-husband's negligence in caring for and providing for their children, and should she leave it to God or resort to the judiciary despite her circumstances?

The maintenance and housing of children are obligatory upon their father according to what is customary. The Sharia judge determines this amount. We advise you to seek the mediation of religious people to counsel your husband on the obligation of spending on his children and to clarify the sin of neglecting them. If that does not work, then you have no option but to refer the matter to the judiciary or to be patient.