D'oh, didn't see the NRE part.
Anywhoo, if you enlist in the Navy without a degree, the only commissioning option available to you will be STA-21. Your recruiter claims that you can apply to the Naval Academy -- and he might be right by accident -- but do you meet all of their requirements? Do you have a LOR from your Congressman? How do you plan to do the interview when you're stuck in Charleston for nuke training or attached to a ship? There's more to getting an appointment to USNA than just filling out a form and crossing your fingers. I'm not pointing this out to discourage you from applying to USNA; rather, that your plan to enlist and send an application off somewhere between boot camp/a-school/NNPS is more complicated than your enlisted recruiter might have told you.
There are other factors to consider, too. Yes, NUPOC can be a competitive program (and much moreso for NRE), but oftentimes recruiters will claim a program is harder than it is to discourage applicants from even trying and encourage them to take the 'easy' way out and enlist.
Applying to NUPOC will guarantee that you work in whatever designator the Admiral accepts you into. Applying to the Naval Academy will guarantee that you do NOT work as an NRE or Instructor (USNA only commissions line officers with some exceptions of people who get medically DQ'd after being selected for a community but are still fit for service in another), and you also only get a dream sheet. You might get picked up for nuke, you might not. It depends on the needs of the Navy, and at that point you're at their mercy.
STA-21 is a great deal, and thus is very competitive. You typically need to build up some good evals and show top-notch performance in the Navy to get accepted.