There are many, many different things that fall under the heading of nuclear physics. There is theory and experiment. There is also low-energy nuclear physics (basically nonrelativistic) and high-energy (overlaps with particle physics). Back when I was doing research (grad student and postdoc), I was doing low-energy experimental nuclear physics. The subfield was nuclear stucture (as opposed to reactions), and the subsubfield was gamma-ray spectroscopy. This was at a university lab and then at a DOE lab.
I don't know how much demand there is for nuclear physics in industry. I would guess that most of those jobs would be in nuclear power or at military contractors building or maintaining bombs, but the expertise required might be more like nuclear engineering. With a PhD in nuclear physics you would certainly be (over-)qualified to work in certain jobs related to health and the environment, e.g., surveying and analyzing waste.