Must one make up prayers if they did not know that nocturnal emission necessitates ghusl, or if they performed ghusl and years later remembered the presence of nail polish, and what are the cases that necessitate making up prayers?

There are two issues here:

First: Purity (Tahara) is a condition for the validity of prayer. The fatwa issued is that whoever prayed in a state of major ritual impurity (janaba) while ignorant of the obligation of ghusl (ritual bath) must repeat those prayers. If the number of prayers is large, they should make up what they predominantly believe will clear their conscience.

Second: If the nail polish is thick enough to prevent water from reaching the nails, then the prayers performed with that incomplete ghusl must be repeated. If it does not prevent water from reaching the nails, then the ghusl is valid and the prayers do not need to be repeated.