The fundamental nature of dark matter so far eludes direct detection experiments, but it has left its imprint in the cosmic large-scale structure. Disentangling this imprint requires accurate modelling of structure formation in non-standard dark matter models, careful handling of astrophysical uncertainties and consistent observations in independent cosmological probes. I will review a multi-scale, multi-epoch test of the nature of dark matter combining state-of-the-art modelling with observations of the cosmic microwave background, galaxy clustering (redshift z < 2), the Lyman-alpha forest (2 < z < 5) and the high-redshift (z > 5) galaxy UV luminosity function from the Hubble and Webb Space Telescopes. I will discuss the extent to which cosmological data are more consistent in the presence of ultra-light axion dark matter. I will further discuss prospects for adjudicating the viability of axion models in observations of the galaxy and Milky Way sub-structure distributions in the transformative Vera C. Rubin Observatory.