I'm gonna have to chime in on this one. Having been an enlisted nuke and spending the last 15 months on the LINCOLN as the Auxiliaries Divo, I think I can add some perspective.
First, you won't necessarily go to a small boy first and then to the carrier as a SWO(N). Although that is the preferred way, the navy also sends SWO(N)'s to the carrier for three year tours, forgoing the small boy experience.
Second, as Wink said, the hard science degree isn't a hard and fast rule. Plenty of liberal arts majors make it in the nuke community as well. The full year of calculus and calc based physics IS, however, a hard and fast rule. For example, even though I have been through the entire curriculum as an enlisted electrician and even graduating in the top 1/3 of my nuke power school class, Naval Reactors refused to allow me to switch over to SWO(N) (from regular SWO after I qualified) because I did not take the full year of Calc and Calc based Physics in college. They simply would not budge.
One thing is definately for certain. If you don't interview....you will have no worries about being selected.
There was a comment about the SWO(N)'s being primarily limited to the engineering spaces making it difficult to get their quals. This has some degree of truth to it. As I said before, the preferred method of qual progression is to go to a small boy first to qualify as a SWO and then go to power school and prototype and then ultimately, the carrier. Your tour on the carrier would be your second divo tour since you had already completed one on the small boy. Having already qualified as a SWO, you should no doubt expect to spend an absurd amount of time down in the main spaces so you can qualify as Propulsion Plant Watch Officer (PPWO). Watchbills in the nuke world are manpower intensive....for a reason. Now, if you show up to the carrier for the three year tour without having first done a first tour divo job on a small boy, you will have to earn both your SWO qual and PPWO right there on the carrier. Because the watchbills take up so much manpower, you may find yourself on two different watchbills (Nuke and Bridge). Depending where the carrier is in its operational cycle, the qual progress for this individual can get interesting.
As for being a career killer if you went SWO(N). Not hardly!
There was a comment in there about "The rest of your career can be on small boys." If you continue your career path in the SWO(N) commnunity, you will complete your tour on the CVN and then head off to shore duty (or go straight to Department Head school for the hard chargers). Then you'll start the Department Head (DH) pipeline and will go through DH school and then to some DH job (there's a plethora of them out there). Most likely you'd be sent to a small boy and you'll be assigned a top sider billet (i.e. OPS, WEPS, CSO). What you won't do if you're a nuke is be the CHENG of a small boy. After you complete your first DH job, off you'll go back to the CVN for your second DH tour filling some nuke billet.
Lastly, if you do decided to to leave the navy after your committment is up, you will not be sought after 10X that of a regular SWO, I'm sorry to say. And you won't be paid twice as much. While you will be more competitive based on the technical training you received as a nuke, you will only command about a 20% higher salary than a regular SWO will. As for the SWO(N) working his/her ass off. Yes that is true. But you will work your ass off as a regular SWO too...guarantee it.