We present N-body and hydrodynamical simulations of the response of the Milky Way's baryonic disc to the presence of the Large Magellanic Cloud during a first infall scenario. For a fiducial Galactic model reproducing the gross properties of the Galaxy, we explore a set of six initial conditions for the Large Magellanic Cloud (LMC) of varying mass which all evolve to fit the measured constraints on its current position and velocity with respect to the Galactic Centre. We find that the LMC can produce strong disturbances - warping of the stellar and gaseous discs - in the Galaxy, without violating constraints from the phase-space distribution of stars in the Solar Neighbourhood. All models correctly reproduce the phases of the warp and its antisymmetrical shape about the disc's mid-plane. If the warp is due to the LMC alone, then the largest mass model is favoured (2.5 x 10(11) M-circle dot). Still, some quantitative discrepancies remain, including deficits in height of Delta Z = 0.7 kpc at R = 22 kpc and Delta Z = 0.7 kpc at R = 16 kpc. This suggests that even higher infall masses for the LMC's halo are allowed by the data. A comparison with the vertical perturbations induced by a heavy Sagittarius dSph model (10(11) M-circle dot) suggest that positive interference with the LMC warp is expected at R = 16 kpc. We conclude that the vertical structure of the Galactic disc beyond the Solar Neighbourhood may jointly be shaped by its most massive satellites. As such, the current structure of the Milky Way suggests we are seeing the process of disc heating by satellite interactions in action.