Apathy is to define as a "… lack of interest, enthusiasm or concern" (Oxford Dict.) like indifference too. Of course. But I see noticeable differences that govern the mental disposition behind those attitudes:
Apathy is very often a matter of mental impairment or disease. It is the lack of motivation or mental energy to participate in anything due to illness like dementia or autism.
Unlike apathy, ignorance is the deliberate decision not to take any notice of something or someone. It can have positive AND negative connotations. In both cases, someone doesn't take care for someone/something, because he/she is not being able or willing to take care for someone/something. He or she could but doesn't want to do it!
Apathy and indifference are synonyms.
Ignorant means to not know (something). a synonym for this word is oblivious.
I don’t know that much about nihilism but it is basically an ideology that “nothing really matters” and “existence is meaningless” kind of thing. (Although it’s worth it to point out that someone can have a “nihilistic” mentality without being pessimistic about it)