Increasing wind and solar electricity generation in power systems increases temporal variability in electricity prices which incentivizes the development of flexible processes for electricity generation and electricity-based fuels/chemicals production. Here, we develop a computational framework for the integrated design and optimization of multi-product processes interacting with the grid under time-varying electricity prices. Our analysis focuses on the case study of nuclear-based hydrogen (H2) and electricity generation, involving nuclear power plants (NPP) producing high temperature heat and electricity coupled with a high temperature steam electrolyzers (HTSE) for H2 production. The ability to co-produce H2 along with nuclear is widely seen as critical to improving the economics of nuclear energy technologies. To that end, our model focuses on evaluating the least-cost design and operations of the NPP-HTSE system while accounting for: a) power consumption variation with current density for the HTSE and the associated capital and operating cost trade-off, b) heat integration between NPP and HTSE and c) temporal variability in electricity prices and their impact on plant operations to meet a baseload hydrogen demand. Instead of formulating a monolithic optimization model, which would be computationally expensive, we propose a decomposition approach that reformulates the original problem into three sub-problems solved in an iterative manner to find near-optimal solutions. Through a numerical case study, we demonstrate the potential synergies of NPP and HTSE integration under alternative electricity price scenarios. This synergy is measured via the metric of relative breakeven H2 selling price that accounts for the opportunity cost of reduced electricity sales from H2 co-production.