We present the first results of the Galaxy Activity, Torus and Outflow Survey (GATOS), a project aimed at understanding the properties of the dusty molecular tori and their connection to the host galaxy in nearby Seyfert galaxies. Our project expands the range of AGN luminosities and Eddington ratios covered by previous surveys of Seyferts conducted by ALMA and allows us to study the gas feeding and feedback cycle in a combined sample of 19 Seyferts. We used ALMA to obtain new images of the emission of molecular gas and dust using the CO(3-2) and HCO+(4-3) lines as well as their underlying continuum emission at 870 microns with high spatial resolutions (0.1 ~ 7 - 13 pc) in the CND of 10 nearby (D < 28 Mpc) Seyfert galaxies. Our new ALMA observations detect 870 micron continuum and CO line emission from spatially resolved disks located around the AGN in all the sources. The bulk of the continuum flux can be accounted for by thermal emission from dust in the majority of the targets. For most of the sources the disks show a preponderant orientation perpendicular to the AGN wind axes, as expected for dusty molecular tori. The median diameters and molecular gas masses of the tori are ~ 42 pc, and ~ 6 x 10**5 Msun, respectively. We find a positive correlation between the line-of-sight gas column densities responsible for the absorption of X-rays and the molecular gas column densities derived from CO towards the AGN in our sources. The radial distributions of molecular gas in the CND of our combined sample show signs of nuclear-scale molecular gas deficits. We also detect molecular outflows in the sources that show the most extreme nuclear-scale gas deficits in our sample. These observations find for the first time supporting evidence that the imprint of AGN feedback is more extreme in higher luminosity and/or higher Eddington ratio Seyfert galaxies.