Is it permissible for a wife to work in nursing, seeing the 'awrah (private parts) of elderly men and helping them with ghusl (ritual bathing), considering this a humanitarian act?

In Fatwa no. 39591, it was previously affirmed that this act is forbidden. Therefore, your wife must abandon it. There is no necessity for her to view private parts, as this can be done by the wife of the incapacitated person, or his male relatives, or female mahrams if they are unavailable. Her claim that they are like children and do not possess instinct is incorrect, and even if it were true, her own instinct is present.