What is one's duty towards a sick mother, children, and husband, given the long distance, difficulty of travel, and the questioner's illness, and which of them has priority in care: the children or the mother?

Thank you for your eagerness to honor your mother and earn her رضا (pleasure/satisfaction). If your mother is under the full care of your father, your siblings, and their wives, then she is not in need of you. There is no blame upon you for staying in your home and visiting her as often as is feasible. If you can convince your husband to let you stay overnight with her, that is good, and he should permit you if possible. If he does not permit you, then your obedience to your husband takes precedence over your obedience to your parents, for her husband possesses her more than her parents, and obedience to him is more obligatory. As long as your mother is under full care, your duty is to care for your children and visit your mother as often as is feasible.