We present first results of cold cosmic gas evolution obtained through a set of state-of-the-art numerical simulations (ColdSIM). We model time-dependent atomic and molecular non-equilibrium chemistry coupled to hydrodynamics, star formation, feedback effects, various UV backgrounds as suggested by the recent literature, HI and H2 self-shielding, H2 dust grain catalysis, photoelectric heating and cosmic-ray heating. By means of such nonequilibriumcalculations we are finally able to reproduce the latest HI and H2 observational data. Consistently with available determinations, neutral-gas mass density parameter results around Ωneutral ∼ 10−3 and increases from lower to higher redshift (z). The molecular-gas mass density parameter shows peak values of ΩH2 ∼ 10−4, while expected H2 fractions can be as high as 50% of the cold gas mass at z ∼ 4-8, in line with the latest measurements from high-z galaxies. These values agree with observations up to z ∼ 7 and both HI and H2 trends are well reproduced by our non-equilibrium H2-based star formation modelling. Corresponding H2 depletion times remain below the Hubble time and comparable to the dynamical time at all epochs. This implies that non-equilibrium molecular cooling is efficient at driving cold-gas collapse in a variety of environments and since the first half Gyr. Our findings suggest that, besides HI, non-equilibrium H2 analyses are key probes for assessing cold gas and the role of UV background radiation.