Is my mother's action correct, and how should she spend on my deceased brother's children, knowing that she spends from her own money, saves the fixed income she receives, and spends the non-fixed income on food, drink, and bills?

What some people give to children becomes their property. Their guardian should spend from it appropriately and save any excess. As for what is given to the family, it belongs to whomever the giver intended with their gift, for the intent matters, not the words. If the mother is needy, she may eat with the children from their money. If she is wealthy, she should refrain and eat from her own money. Allah says: "And whoever is rich, let him abstain [from taking a fee]; and whoever is poor, let him eat in a reasonable amount."