UK Government projections anticipate increasing electricity use, provided by variable renewables (i.e., wind and solar PV). A side effect of increasing the proportion of variable renewable generation is increased support costs, including curtailment, energy storage, and (most significantly) the cost of supplying electricity for periods of high demand when variable renewable generation is low. As the proportion of variable renewable capacity increases, demand for supporting capacity increases but the capacity factor of the support generation decreases, raising the support costs. Using nuclear power for dedicated baseload supply makes the situation worse. This paper explores in the UK context an original low-cost solution using nuclear cogeneration with hydrogen production as the main application. Electricity is diverted at low cost to the grid at times of high demand when renewables are not available. This ensures nuclear maintains a high capacity factor. When higher temperature advanced systems become available, using thermal energy storage will increase the nuclear electrical capacity. This “Flexible Nuclear” scenario substantially reduces support costs for accommodating variable renewables, saving GBP 14 bn/yr and leading to an 80% reduction in CO2 equivalent emissions, compared to a recent UK Government scenario utilising a large capacity of hydrogen and unabated gas generation at very low capacity factors.