Accurately predicting how the cosmic abundance of neutral hydrogen evolves with redshift is a challenging problem facing modellers of galaxy formation. We investigate the predictions of four currently favoured semi-analytical galaxy formation models applied to the Millennium simulation for the mass function of cold neutral gas (atomic and molecular) in galaxies as a function of redshift, and we use these predictions to construct number counts for the next generation of all-sky neutral atomic hydrogen (H I) surveys. Despite the different implementations of the physical ingredients of galaxy formation, we find that the model predictions are broadly consistent with one another; the key differences reflect how the models treat active galactic nuclei feedback and how the time-scale for star formation evolves with redshift. The models produce mass functions of cold gas in galaxies that are generally in good agreement with H I surveys at Graphic. Interestingly, we find that these mass functions do not evolve significantly with redshift. Adopting a simple conversion factor for cold gas mass to H I mass that we apply to all galaxies at all redshifts, we derive mass functions of H I in galaxies from the predicted mass functions of cold gas, which we use to predict the number counts of sources likely to be detected by H I surveys on next generation radio telescopes such as the Square Kilometre Array and its pathfinders. We find the number counts peak at Graphic galaxies degGraphic at Graphic for a year long H I hemispheric survey on a 1/10/100 per cent SKA with a 30 degGraphic field of view, corresponding to an integration time of 12 h. On a full SKA with a 200 degGraphic field of view (equivalent to an integration time of 80 h) the number counts peak at Graphic galaxies degGraphic at Graphic. We show also how adopting a conversion factor for cold gas mass to H I mass that varies from galaxy to galaxy impacts on number counts. In addition, we examine how the typical angular sizes of galaxies vary with redshift. These decline strongly with increasing redshift at Graphic and more gently at Graphic; the median angular size varies between 5 and 10 arcsec at Graphic, 0.5 and 3 arcsec at Graphic and 0.2 and 1 arcsec at Graphic for galaxies with H I masses in excess of Graphic, depending on the precise model. Taken together, these results make clear that the forthcoming H I surveys will provide important and powerful tests of theoretical galaxy formation models.