Good advice in this thread. One minor thing to clear up: NUPOC is how all civilian to nuke OCS guys enter the Navy, even with a degree. You collect like 1-2 months of E-6 pay while you wait to class up, and everyone gets jealous that you collect BAH while at OCS because it's TAD orders and you are permanently "stationed" at your NRD.
OP, if you want to become eligible for the nuke program ASAP, then go back to school for a year and take a calculus and calculus-based physics sequence, and get straight A's. That
may qualify you for the program, but ask an officer recruiter before you spend money on post-bacchelorate classes -- the summary the interviewers and ADM at NR see have your technical grades, standardized test grades (e.g. SAT), and your overall GPA.
If you want a full-fledged engineering career, the only path to that lies in getting a full-fledged undergraduate engineering degree from an ABET accredited university. The Navy will not enable you to do that while you are active duty, regardless of whether you are an Officer or enlist, and since you already have an undergraduate degree there are limited programs to pay for a second one if you get commissioned. There are programs for you to get graduate degrees via NPS, NWC, or USNA, but they all will require a DH tour commitment (i.e. another sea tour) and you will not be able to pursue one in an engineering/technical course study because of your undergraduate degree.
If you go pilot you can consider these graduate programs to not exist at all because your shore duty performance will matter to the O-4/DH selection board (it occurs while you are on your first shore duty vice when you are at sea doing your DH tour for SWO/subs), so you will be competing for a flight instructor job unless your CO writes that you poop rainbows and gives you EPs on all your fitreps at sea.
It also means the nuclear community is getting smaller in the navy, and therefore more selective - which in your case may hurt you. Lastly (what I talked about above) if you were planning on going navy nuke to prep for a civilian nuke position, the timing may not be in your favor. However, going through the navy nuke program will still provide you with good training and experience that you can carry into other industries.
There are no plans to reduce the size of the carrier or submarine fleets at this moment, which will keep the amount of nuke billets available relatively constant. The VA class contract goes out to the 2020s and possibly beyond; there's no turning back on that unless Uncle Sam wants to pay EB for not building submarines. Judging by the success of this acquisition program among dozens of glaring failures paired with the operational utility of the fast attack fleet, I don't see that happening.
Selectivity is tougher, so I hear, but that's because unemployment is high and stagnant, which makes people turn toward the military for additional skills and OJT to add to their resumes. On the sub officer end, 1-2 year groups ago had very high retention and the detailers project this trend will continue for a few more years. I don't have fleet numbers but usedtafish had an unusually high amount of enlisted nukes report aboard with college degrees and in their mid 20s; these are people who enlisted in '09-'11 because they couldn't find employment and in some cases couldn't financially wait 6-12 months for an OCS application to be processed. I suspect that this trend is not unique to usedtafish.
The military and DOD turned off things it can easily turn back on, like delaying maintenance availabilites, invoking furloughs, and telling ships to stay parked next to the pier. None of this saves a huge amount of money in the scheme of things, but it is the most easily adjustable aspect of the budget -- Operations and Maintenance. What could happen is lowering bonuses, but there has been no official word about that.