I almost didn't see your post since it was in the Marine Corps area. I'm in the NUPOC program, so I can answer any specific questions you have if you send me a Private Message (PM).
Here is some basic info though. Since you are starting your junior year, I'm assuming you have completed your calculus and physics requirements (depending on your major), so you are good to go as far as applying for the program. If accepted, the Navy will pay you somewhere around $2500 a month to complete your degree. Once finished, you go to OCS for commissioning. Then you go to Nuke power school for 24 weeks, then prototype training where you get hands on reactor experience (also 24 weeks). After this, you either go to submarine officer school or surface school depending on whether you volunteer for submarine duty like myself or go for the aircraft carriers. That is your last step before getting your first assignment in the fleet.
As far as the application process goes, you have to do a Physical Readiness Test (PRT), physical exam, security background check, and a regular application. You also need 3 references and all your transcripts. A competitive GPA is around 3.3/4. After submitting the app and completing the physical, you go on a VIP trip to tour a submarine or surface ship to make sure this is what you want to do. If you make it through initial screening, you are given an interview in Washington D.C. At the interview, you basically do two things. You interview with a minimum of two Naval Reactor engineers, who ask you questions about calculus, physics, and classes in your major. After this, you have an interview with Admiral Bowman, the head of the Nuclear Propulsion Program. Within minutes of completing the interview with the Admiral, you find out if you are accepted to the program or not, and if you are, you go back to the hotel and swear in.
The committment is 5 years from the date you are commissioned, not from when you swear in. The good thing is that the time you spend on active duty while in college gets credited to you, so you are earning leave and service time (which helps with your pay later). By the time you finish college you could have more than a month of leave time earned already. You also get the nice $10,000 signing bonus right away, plus $2,000 when you complete the prototype training. After your initial committment, I believe the yearly signing bonus is $22,000.