No kidding. The post-9/11 GI Bill covered my full tuition in law school.
Anyway, to the OP, there is nothing remarkable about signing on to the nuke program. In fact, we were just discussing (as a bit of a hijack) how much recruiters love to pressure qualified candidates into the program.
So my biggest worry wouldn’t be that the recruiter lied to him about being guaranteed advanced nuke training to trick him into joining the Navy. It would be that the recruiter lied to him about how awesome it is to be a nuke to trick him into filling a hard-to-fill quota for the nuke program.
Anyway, for future reference for anyone with family with similar aims, the trick is to apply for an ROTC scholarship concurrent with applying to an academy. Otherwise, by the time you get denied by the Academy, you’re probably too late to apply for an ROTC scholarship for the coming year, and would have to find some other way to cover the cost of tuition for the first year at least—and then there’s no guarantee you’d get picked up for a scholarship later.