The National Labor Relations Board recently determined that scholarship college athletes at private nonprofit colleges and universities can unionize. The NLRB determined that college athletes at those schools are "employees" of the school, and their payment is their scholarship.
Under the National Labor Relations Act, states are not "employers." Since public schools are part of their state government, public schools are not employers under the NLRA. Since private schools are not part of their state government, they are not employers under the NLRA. The federal regulations of the NLRB explicitly provide that the NLRB will exercise jurisdictions over issues involving private nonprofit colleges and universities that have a gross annual revenue of $1M or more.
This does not mean that college athletes at public schools cannot unionize, it just means they have to do so differently. Those athletes would have to unionize under procedures for public sector employees. Some states are right to work states, which prevents compulsory union participation for public sector employees. Those employees could join the union if they wanted to do so, but they would not have to join or participate in its labor policies or goals.
On the other hand, Northwestern's college athletes will be voting within the month to determine whether they will be represented by the College Athletes Players Association. The CAPA was just recently formed. If a majority of Northwestern's athletes vote for representation, then all Northwestern athletes will be represented by the CAPA.