Actually, the question is what the parent, as a proxy for the child, who is the hospitalized parent’s next-of-kin, sort of, should be told.
I think legally, a minor is not next-of-kin, unless the minor has been specifically designated as such (a person who is ill and could become unconscious, and has a 16 1/2 year old might have some kind of power of attorney in place that allows the child to be considered an adult for purposes of discussing the parent’s medical information with the doctor-- just getting information, not making decisions).
If the parent who is not the one who is ill is not sure who is the next of kin, he might ask, because that would be the person to discuss the parent’s situation with the child. If the parent is that out-of-touch with the ex’s family, I don’t think it would be unreasonable to ask the doctor to please have a family member call ASAP so someone can tell the child what is going on.
I understand problems of confidentiality, but I also think it’s much worse for the child to be left in the dark as to the other parent’s condition.