In this work an ensemble of simulated Local Group analogues is used to constrain the properties of the mass assembly history of the Milky Way (MW) and Andromeda (M31) galaxies. These objects have been obtained using the constrained simulation technique, which ensures that simulated LGs live within a large scale environment akin to the observed one. Our results are compared against a standard Λ Cold Dark Matter (ΛCDM) series of simulations which use the same cosmological parameters. This allows us to single out the effects of the constraints on the results. We find that (a) the median constrained merging histories for M31 and MW live above the standard ones at the 1-σ level, (b) the median formation time takes place ≈0.5 Gyr earlier than unconstrained values, while the latest major merger happens on average 1.5 Gyr earlier and (c) the probability for both LG haloes to have experienced their last major merger in the first half of the history of the Universe is ≈ 50% higher for the constrained pairs. These results have been estimated to be significant at the 99% confidence level by means of a Kolmogorov-Simirnov test. These results suggest that the particular environment in which the Milky Way and Andromeda form plays a role in shaping their properties, and favours earlier formation and last major merger time values in agreement with other observational and theoretical considerations.