The people who do this testing (WIAT etc called educational evaluations) can be neuropsychologists, educational consultants or others. The wait for private evaluations in Northern NJ can be quite long - sometimes 2 months. Your child study team or GC will be able to give you some names. We used the Essex County team many years ago and they were great. (If in private school, it will be the county where the school is located who does your testing but your school will have to agree it is necessary and arrange it for you). If your school is not being cooperative, you can arrange for private testing. We also used Phyllis Lakin in Denville. Here are some resources:
http://www.morrispace.org/id17.html