Just to be clear, Naval Reactors Engineers at NR headquarters in DC don't actually do any of the design or research themselves. They are more of a liaison between the labs, contractors, fleet, and shipyards providing the government oversight and approval of the entire process. They need to have a very strong engineering background in order to understand the concepts behind the design and operation of the reactors, but the actual research/design work is done by civilians, not military officers.
Also, for NUPOC, you need to be within 18 months of graduation for a bachelor's, or 12 months of graduation for a master's to apply for the NRE position. So, while you could do a 4+1 program, you would have to wait until the last year before starting your application. If you are interested in Subs/SWO-N/Instructor, you can apply as early as 30 months from graduation for a bachelor's or 12 months for a master's.
You also need to make sure to keep your grades up throughout college, as NR likes to see a 3.8+ GPA for the NRE candidates, and even then they are still very selective on who is invited to interview for the job.