Possibly, but mostly people applying for nuclear engineering have engineering degrees, simply because they would usually have some relevant experience in electrical engineering (electronics, digital instruments, etc), or mechanical engineering (heat transfer, fluid mechanics. power conversion), materials science/engineering (materials performance) and so on.
What electives did one have in addition to the core physics program? Any applied physics?
This is pretty common. If you are wanting to be a reactor operator or something, it may be advantageous for you to go back and get an NE BS simply because it's the easiest path to get a professional engineering license. Otherwise, applying for MS in N.E. programs would be a good idea.