As you have seen here on SC cutting isn't a phase. It can last through to adulthood. I can't believe a professional gave her such non-sense for advice. Some kids do this to fit in with their peers but that just shows they are struggling as well. I have seen many kids who do this and none of them were phases. It really makes me angry that someone would tell her that.
Telling the school could very well backfire for your neice. If she doesn't have anyone treating her as a normal individual she may always feel as if something is wrong with her. If there is a threat of suicide then yes the school should be notified. I don't think cutting has to do with suicide. It is used primarily as a way to survive, to feel, and to cope. Your neice needs to know that help is there if she chooses to take it though.
I hope her mother sees through this persons ignorance and seeks another opinion. A trained psychiatrist who specializes in self-mutilation would be preferred.