What is the ruling on applying nail polish for medical or protective reasons, such as when a woman suffering from cancer is compelled to paint her nails with a type of polish to protect them from breaking and splitting due to chemotherapy, and is it permissible for her to wipe over the polish during ablution given the long duration of the treatment?

If the polish is a treatment for nail shedding, then there is no objection to using it for the duration of the need for treatment. It is not required to remove it for ablution (wudu') or ritual bathing (ghusl (definition)), and its ruling is like that of a splint (jabeera) which is not removed if harm is feared. It is sufficient for water to flow over the polish, and the duration is not limited to a specific time.