To respond to CaptMJ - yes, there are people who do not pass the interviews. My son is a MIDN 1/C who was selected for surface nuke; his buddy is in subs. Just to clarify with very current (NROTC) info - and hopefully help anyone going nuke:
1. Over the summer between your junior and senior years you will put in your top 5 requests for warfare community (SEALs, subs, aviation, surface, etc). Generally, at least one of the top three must be a nuclear billet (surface nuclear or subs), and one must be a restricted line position (Seabees or info warfare, for example, usually listed fifth). The deadline to submit requests this past year was early August. Your NROTC unit will coordinate this. By the way, there is some "gamesmenship" to the ordering of your selections; my son certainly didn't want surface nuke and probably would have gotten aviation if he had listed it in his top three, but he wants to lead sailors and there aren't many opportunities for leadership in aviation for the first eight or so years of your career.
2. Warfare community assignments were announced this year in early November. The info is provided to your unit, and they decide when to tell you (usually, right away).
3. All MIDN selected for nuke billets (sub and surface) are required to have an "interview" with the admiral who is the head of Navy Nuclear Propulsion (and who is also an Assistant Secretary of Energy) in DC. These are generally done in January and February. The interview consists of several separate lengthy interviews with panels of junior officers/Navy civilians and a brief one-on-one meeting with the admiral. The junior officer panels write up an assessment which they provide to the admiral before you see him.
4. If they're on the ball, your unit will prep you EXTENSIVELY for these interviews. People do fail, and failing looks bad for the unit. Of the 25 MIDN interviewing on the day my son did, four failed. You know that day - there's no mystery.
5. The interview questions they can ask vary greatly. For my son, who goes to a very top engineering school, had good grades and didn't really want to be nuke, they asked him very basic questions that required nothing more than 7th grade algebra and physics - i.e., they weren't going to give him any chance to fail (as I think someone correctly noted above, there are a lot of billets for nuke officers and they are hard to fill with qualified candidates). Others, they asked questions that required knowledge of differential equations and thermodynamics. What they really want to see is if you have the skills to pass nuke school, and the mindset to work very hard and very long hours running a reactor. For some people, even engineers, nuke school is the hardest thing they will ever do; for others, it's a joke - it just depends on the preparation you've had going in. Engineering curriculums that focus on a lot of theory seem to be the best prep, but that's just my conjecture.
6. Failing the interview generally means you're going to be reassigned as a conventional SWO - although the "penalty" seems to be that you drop down on the order of merit list for ship selection (if they get you realigned in time for that at all...), so you might be looking at a "leftover" selection (e.g., an amphib out of Norfolk - not that there's anything wrong with that - you're just not going to get a destroyer out of Pearl Harbor or Spain).
It's important to note that the Navy operates about as many nuclear reactors as all of the commercial power plants in the United States combined. The reactors never shut down, whether you're at sea or in port, so nuke officers generally have the longest hours of anyone in the Navy - part of the reason they get paid more. In addition, a nuclear incident aboard a Navy ship or sub would be a catastrophe - there's nowhere to evacuate the crew, you can't hide from radiation, and no one would want that ship pulling into their port (and can you imagine Congress approving a new nuclear carrier because, gee, the last one melted down....?). As a result, the Navy can't afford to make any nuclear mistakes.
With graduation coming in May, this is my swan song from the board. Special thanks to Kinnem for his many thoughtful posts over the past four plus years, as my son went College Program and through the sideload scholarship process.