Is your child going to kindergarten at the public school or the private school? My personal experience in dealing with a private school and getting services through the district while you are in the private school was not a good one.
While the private school can help you with this, they are NOT required to do it. Your school district is required, but they might want a doctor's recommendation to do an evaluation.
What about the place you got the evaluation? Can they give you any guidelines? Can you get a letter to help move the evaluation process along? Then you could tell the district you have a letter saying that services are required.
You need to move fast on this as the end of the school year isn't that far away and they do not like to do evaluations when school is not in session.
M.