My daughter is in a private lutheran school in Queens and they cater to a lot of international students. I had no problem enrolling her for nursery mid school year and they explained to me that some of their students are only there for a few months out of the year since parents have to travel or work in NYC only a few months and then work the remaining time abroad.
If interested in a school in the Rego Park section of Queens, I would recommend Our Saviour Lutheran School. They are great. They have half days, full days and extended days with early and late hours available. If not, I would probably recommend looking into private schools as they are more accomodating than public.
Good luck with your search.