  (Abridged) 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 (HI) 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 AGN feedback and how
the timescale for star formation evolves with redshift. The models produce mass
functions of cold gas in galaxies that are generally in good agreement with HI
surveys at $z$=0. Interestingly we find that these mass functions do not evolve
significantly with redshift. Adopting a simple conversion factor for cold gas
mass to HI mass that we apply to all galaxies at all redshifts, we derive mass
functions of HI 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
HI surveys on next generation radio telescopes such as the Square Kilometre
Array and its pathfinders. We show also how adopting a conversion factor for
cold gas mass to HI 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. Taken together, these results make clear that forthcoming HI
surveys will provide important and powerful tests of theoretical galaxy
formation models.
