Nuclear assisted low carbon hydrogen production by water electrolysis represents a potential application of nuclear cogeneration towards deep decarbonization of several fossil fuel-dependent industrial sectors. This work builds a probabilistic techno-commercial model of a water electrolysis plant coupled to an existing nuclear reactor for base load operations. The objective is to perform discounted cash flow (DCF) calculations for levelized nuclear hydrogen production cost under input parameter uncertainty. The probability distributions of inputs are used with the Monte Carlo-Latin Hypercube (MC-LH) sampling technique to generate 10⁵ input scenarios and corresponding distribution of the levelized or life cycle hydrogen production cost instead of deterministic point values. Based on current techno-economic conditions, the levelized production costs of electrolytic hydrogen using electricity from large water-cooled nuclear reactors are determined to be US $ 12.205 ± 1.342, 8.384 ± 1.148 and 6.385 ± 1.051/kg H₂ respectively at rated alkaline water electrolyser capacities of 1.25 MW(e), 2.5 MW(e) and 5 MW(e). The corresponding values for PEM water electrolysers are US $ 13.162 ± 1.356, 8.891 ± 1.141 and 6.663 ± 1.057/kg H₂. The potential for flexible nuclear reactor operation and management of power demand uncertainties through nuclear hydrogen cogeneration is also examined through a case study.