First piece of advice; lose the attitude about your education being better than public school and the government making you take another test because you didn't attend a "government school".
Sections 2.d, 2.e, and 2.f of the below link all specifically mention high school.
http://www.public.navy.mil/bupers-npc/reference/milpersman/1000/1300Assignment/Documents/1306-502.pdfAdditionally, the recruiters have their own procedures that they must follow, with specific instructions for home schooled individuals.
As a Nuke, regulations, procedures and established processes are simply not put in place so the government has control of stuff ( for the most part, that is). Our entire field (Navy) and industry (civilian) revolve around the simple process of procedural compliance. We do NOT deviate from approved procedures and processes without good reason. When we do deviate, there is a process laid out to do just that while rectifying the deficiency that drove us in that direction in the first place.
Additionally, neither high ASVAB nor Nuke field test scores are any guarantee of success in the Nuke field. I had a guy in my class score a 99, along with a 78 on the Nuke test. He also was Valedictorian of his private high school graduating class, which he took every opportunity to remind us "public school flunkies" (his words) of. He failed out week 7 of Power School.
Both tests, coupled with exposure to chemistry, algebra, and physics from high school, simply give a rough indicator of an individual's
potential to learn and understand the concepts taught and successfully complete the training pipeline.
Simply put, you have the potential to complete the Navy's Nuclear training program. If you can handle the military side of things as well, that is.