The ability of high-temperature solid oxide cell (SOC) electrochemical reactors to efficiently convert atmospheric carbon to high value chemicals for industrial and energy storage applications via CO2 and co-electrolysis makes them a key technology for active carbon utilisation. However, due to additional operational risks from thermochemical reactions on thermal management, limited experimental capacity, and relative novelty, CO2 and co-electrolysis lag behind steam electrolysis in large-scale adoption. Here, a 1D+1D SOC model based on fundamental first principles considering three-dimensional heat transfer was improved via a unique method for representing co-electrolysis electrochemistry, solving with low computational effort. Validation against experimental data for two compositions and pressures, showed high levels of accuracy with respect to characteristic cell voltages, temperatures, and outlet compositions. The model also showed CO2 reduction during co-electrolysis mainly occurred via reverse water gas shift, while CO2 electrolysis still accounted for up to 35% of the total share. Pressurised co-electrolysis operation promotes exothermic methanation, causing pronounced heating of the reactor, consequently reducing the isothermal current density. Therefore, low to moderate pressurisation is likely most suited for coupling with downstream synthesis processes to avoid the installation of unnecessarily large systems and associated high costs.