Hydrogen is known to be the carbon-neutral alternative energy carrier with the highest energy density. Currently, more than 95% of hydrogen production technologies rely on fossil fuels, resulting in greenhouse gas emissions. Water electrolysis is one of the most widely used technologies for hydrogen generation. Nuclear power, a renewable energy source, can provide the heat needed for the process of steam electrolysis for clean hydrogen production. This review paper analyses the recent progress in hydrogen generation via high-temperature steam electrolysis through solid oxide electrolysis cells using nuclear thermal energy. Protons and oxygen-ions conducting solid oxide electrolysis processes are discussed in this paper. The scope of this review report covers a broad range, including the recent advances in material development for each component (i.e., hydrogen electrode, oxygen electrode, electrolyte, interconnect, and sealant), degradation mechanisms, and countermeasures to mitigate them.