We examine the spatial distribution of the oldest and most metal-poor stellar populations of Milky Way-sized galaxies using the A Project Of Simulating The Local Environment (APOSTLE) cosmological hydrodynamical simulations of the Local Group. In agreement with earlier work, we find strong radial gradients in the fraction of the oldest (t(form) <0.8 Gyr) and most metal-poor ([Fe/H] <-2.5) stars, both of which increase outwards. The most metal-poor stars form over an extended period of time; half of them form after z = 5.3, and the last 10 per cent after z = 2.8. The age of the metal-poor stellar population also shows significant variation with environment; a high fraction of them are old in the galaxy's central regions and an even higher fraction in some individual dwarf galaxies, with substantial scatter from dwarf to dwarf. We investigate the dependence of these results on the assumptions made for metal mixing. Overall, over half of the stars that belong to both the oldest and most metal-poor population are found outside the solar circle. Somewhat counter-intuitively, we find that dwarf galaxies with a large fraction of metal-poor stars that are very old are systems where metal-poor stars are relatively rare, but where a substantial old population is present. Our results provide guidance for interpreting the results of surveys designed to hunt for the earliest and most pristine stellar component of our Milky Way
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