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The nature of the dark matter can affect the collapse time of dark matter haloes, and can therefore be imprinted in observables such as the stellar population ages and star formation histories of dwarf galaxies. In this paper, we use high-resolution hydrodynamical simulations of Local Group-analogue (LG) volumes in cold dark matter (CDM), sterile neutrino warm dark matter (WDM) and self-interacting dark matter (SIDM) models with the eagle galaxy formation code to study how galaxy formation times change with dark matter model. We are able to identify the same haloes in different simulations, since they share the same initial density field phases. We find that the stellar mass of galaxies depends systematically on resolution, and can differ by as much as a factor of 2 in haloes of a given dark matter mass. The evolution of the stellar populations in SIDM is largely identical to that of CDM, but in WDM early star formation is instead suppressed. The time at which LG haloes can begin to form stars through atomic cooling is delayed by ∼200 Myr in WDM models compared to CDM. It will be necessary to measure stellar ages of old populations to a precision of better than 100 Myr, and to address degeneracies with the redshift of reionization - and potentially other baryonic processes - in order to use these observables to distinguish between dark matter models.