We use high-resolution data from the millimetre-Wave Interferometric Survey of Dark Object Masses (WISDOM) project to investigate the connection between circumnuclear gas reservoirs and nuclear activity in a sample of nearby galaxies. Our sample spans a wide range of nuclear activity types including radio galaxies, Seyfert galaxies, low-luminosity active galactic nuclei (AGN) and inactive galaxies. We use measurements of nuclear millimetre continuum emission along with other archival tracers of AGN accretion/activity to investigate previous claims that at, circumnuclear scales (<100 pc), these should correlate with the mass of the cold molecular gas. We find that the molecular gas mass does not correlate with any tracer of nuclear activity. This suggests the level of nuclear activity cannot solely be regulated by the amount of cold gas around the supermassive black hole (SMBH). This indicates that AGN fuelling, that drives gas from the large scale galaxy to the nuclear regions, is not a ubiquitous process and may vary between AGN type, with timescale variations likely to be very important. By studying the structure of the central molecular gas reservoirs, we find our galaxies have a range of nuclear molecular gas concentrations. This could indicate that some of our galaxies may have had their circumnuclear regions impacted by AGN feedback, even though they currently have low nuclear activity. On the other hand, the nuclear molecular gas concentrations in our galaxies could instead be set by secular processes.