I live in a state and city similar to NYC. Nurses can apply by going to the board of education web site where there are posted job openings for teachers, nurses, other health care providers. Many of the large cities will start nurses at the same salaries as the teachers,however, there is usually some kind of school nurse certification required. When I looked into this, my understanding is that the nurses were required to take the same basic school skills test as entry level teachers, no easy task, if you have been away from school for awhile. After test completion, there was additional course work to be completed at a university and clinicals to complete the certification requirement. I hope you get more replies to you post.
I have also come across ads/ from nursing agencies which to not advertise for certifications. I wonder if these are really school nursing jobs, or more like nurse caregiver for a specific patient. You may want to look into this. However, I suspect the pay and benefits are not as good.