In this Roadmap, we present a vision for the future of submillimetre and millimetre astronomy in the United Kingdom over the next decade and beyond. This Roadmap has been developed in response to the recommendation of the Astronomy Advisory Panel (AAP) of the STFC in the AAP Astronomy Roadmap 2022. In order to develop our stragetic priorities and recommendations, we surveyed the UK submillimetre and millimetre community to determine their key priorities for both the near-term and long-term future of the field. We further performed detailed reviews of UK leadership in submillimetre/millimetre science and instrumentation. Our key strategic priorities are as follows: 1. The UK must be a key partner in the forthcoming AtLAST telescope, for which it is essential that the UK remains a key partner in the JCMT in the intermediate term. 2. The UK must maintain, and if possible enhance, access to ALMA and aim to lead parts of instrument development for ALMA2040. Our strategic priorities complement one another: AtLAST (a 50m single-dish telescope) and an upgraded ALMA (a large configurable interferometric array) would be in synergy, not competition, with one another. Both have identified and are working towards the same overarching science goals, and both are required in order to fully address these goals.