Here are the academic requirements:
1. All applicants for the NF program must be High School Diploma Graduates (HSDG) from a state accredited high school. Applicants who attend home school or any other non-traditional program during high school, applicants who attained HSDG status by any means other than a state accredited high school diploma, and applicants that graduated from a foreign high school require a NF Type 2 Non-Traditional Education waiver.
2. NF candidates must provide complete high school and (if attended) college transcripts.
3. Applicants must have completed one year of any high school or college level algebra.
4. Applicants must not have failed any math class (waiverable).
5. Applicants must not have withdrawn from, or repeated, any portion of any math class (waiverable).
6. All non-traditional education applicants must take (and pass) the Navy Advanced Placement Test (NAPT), which is the latest version of the "Nuke Test".
7. All applicants must have qualifying scores on the ASVAB. The overall ASVAB score is not especially important, just 65 or higher, but the combination of specific scores (line scores) is important. There are 2 calculations of line scores that determine eligibility for the NF.
In other words, if your son actually finishes a GED, he will not be eligible for the Navy's Nuclear Field. If he returns to a high school and gets a diploma, or takes classes at a local community college and gets a diploma, it is possible, if he meets all the other requirements, to submit a waiver for his non-traditional education.
To take the ASVAB, he can drop by your local Navy recruiting office. They will get a bunch of personal information from him to verify basic eligibility, and set him up to take the exam. They will also be able to tell him about his limitations wrt his education and joining the Navy in any of our ratings.
Good luck!