We study the mass growth histories of the haloes of Milky Way and M31 analogues formed in constrained cosmological simulations of the Local Group. These gravity-only simulations constitute a fair and representative set of Λ cold dark matter (ΛCDM) realisations conditioned on the masses, positions, and relative velocities of the two big haloes and on the observed recession velocities at the positions of nearby isolated galaxies. Our M31 haloes have similar mass growth histories as the isolated analogues in the TNG dark-matter-only simulations, while our Milky Ways typically form earlier, with suppressed growth at late times. On average, our Milky Ways assemble half their halo mass by ⟨z₅₀⟩ = 1.4 and our M31s by ⟨z₅₀⟩ = 1.2, whereas ⟨z₅₀⟩ = 1.1 for their isolated analogues. Mass growth associated with major and minor mergers is also biased early for the Milky Way in comparison to M31. Most accretion occurs 1–4 Gyr after the Big Bang; growth at later times is relatively quiescent. Based on the mass ratio and time of infall, we find that 32% of our Milky Ways experienced a Gaia-Enceladus-Sausage-like merger, 13% host a massive Large Magellanic Cloud-like satellite at the present day, and 5% have both. In one case, a Small Magellanic Cloud analogue and a Sagittarius analogue are also present, showing that the most important mergers of the Milky Way can be reproduced in its Local Group environment in ΛCDM. We find that the material that makes up the Milky Way and M31 haloes at the present day initially collapsed onto a plane roughly aligned with the Local Sheet and super-galactic plane; after z ∼ 2, accretion occurred mostly within this plane, with the tidal effects of the heavier companion, M31, significantly impacting the late growth history of the Milky Way.