Ruminant livestock produce more than half of the UK’s agricultural greenhouse gas emissions, leading the Climate Change Committee to call for diets to move away from meat and dairy. We run simulations in Defra’s partial equilibrium model of the UK’s agricultural economy to evaluate how diet change might affect UK herd sizes and associated greenhouse gas emissions. We also simulate carbon tax and tariff policy scenarios to compare how diet shifts would interact with a widely advocated policy measure. We find unilateral diet change in the UK alone more likely to provoke a decrease in imports (and potentially an increase in exports) than bring about a significant reduction in UK ruminant herds and associated UK territorial greenhouse gas emissions. Conversely, our simulations find a large carbon tax imposed on domestic farmers alone reducing territorial emissions significantly, but only by leading to higher imports (and associated emissions) from overseas as UK consumption remains inelastic. Our modelling indicates that meeting the UK’s agricultural greenhouse gas mitigation goals requires holistic action on the consumption and production side of the economy, with the UK facing unintended consequences in its agri-food trade balance if its climate ambition is not in harmony with its trade policy.