How can the father’s condition of having to greet his new wife be reconciled with the mother’s condition of not greeting her, knowing that the father abandoned the mother and she has become dependent on her children?

When the commands of the parents conflict, obedience to the mother is prioritized, provided that the command of each of them is in accordance with what is good (ma'ruf). The mother's command not to greet the stepmother is not in accordance with what is good. Therefore, the father must be obeyed by greeting his wife. If it is possible to conceal this from the mother, that would be better. The father should also be advised to be just between his two wives and warned against inclining towards one over the other.