To be a 'nuclear engineer', one would essentially need a degree in nuclear engineering. One possibility would be to obtain an MS degree in nuclear engineering if one already had a baccalaureate in ME.
However, much of 'nuclear engineering' is mechanical engineering - mechanics of materials, thermodynamics, fluid mechanics, corrosion, and materials issues.
The nuclear reactor/core is really where the 'nuclear' part of nuclear engineering is primarily focused, although radiation shielding and radiation sources are another area.
Radiation protection is handled in a parallel discipline - radiation health physics.
Manufacturers of nuclear systems and nuclear fuel employ mechanical engineers and process/chemical engineers to design/develop/manufacture (including supervision) the products (e.g. fuel and core components) and the process used in manufacturing. Utilities would hire ME's in various aspects of plant operation, but those activities related to the fuel and core are generally handled by nuclear engineers.