The decarbonisation of the energy system is crucial for achieving climate goals and is inherently linked to the decarbonisation of industry. Despite this, few studies explore the simultaneous impacts of decarbonising both sectors. This paper aims to examine how industrial decarbonisation in Europe affects the energy system and vice versa. To address this, an industry model incorporating key heavy industry sectors across six European countries is combined with an energy system model for electricity and hydrogen covering fifteen European regions, refered to as the EU-15, divided into eleven zones. The study evaluates various policy scenarios under different conditions. The results demonstrate that industrial decarbonisation leads to a significant increase in electricity and hydrogen demand. This additional demand for electricity is largely met through renewable energy sources, while hydrogen supply is predominantly addressed by blue hydrogen production when fossil fuels are authorized and the system lacks renewable energy. This increased demand results in higher prices with considerable regional disparities. Furthermore, the findings reveal that, regardless of the scenario, the electricity mix in the EU-15 remains predominantly renewable, exceeding 85%. A reduction in carbon taxes lowers the prices of electricity and hydrogen, but does not increase consumption, as the lower carbon tax makes the continued use of fossil fuels more attractive to industry. In scenarios that enforce a phase-out of fossil fuels, electricity prices rise, leading to a greater reliance on imports of low-carbon hydrogen and methanol. Results also suggest that domestic hydrogen production benefits from synergies between electrolytic hydrogen and blue hydrogen, helping to maintain competitive prices.