We use the combined data sets of the Millennium I and II cosmological simulations to revisit the impact of mergers in the growth of bulges in central galaxies in the Λ cold dark matter (ΛCDM) scenario. We seed galaxies within the growing CDM haloes using semi-empirical relations to assign stellar and gaseous masses, and an analytic treatment to estimate the transfer of stellar mass to the bulge of the remnant after a galaxy merger. We find that this model roughly reproduces the observed correlation between the bulge-to-total mass (B/T) ratio and stellar mass (M*) in present-day central galaxies as well as their observed demographics, although low-mass B/T < 0.1 (bulgeless) galaxies might be scarce relative to the observed abundance. In our merger-driven scenario, bulges have a composite stellar population made of (i) stars acquired from infalling satellites, (ii) stars transferred from the primary disc due to merger-induced perturbations and (iii) newly formed stars in starbursts triggered by mergers. We find that the first two are the main channels of mass assembly, with the first one being dominant for massive galaxies, creating large bulges with different stellar populations than those of the inner discs, while the second is dominant for intermediate/low-mass galaxies and creates small bulges with similar stellar populations to the inner discs. We associate the dominion of the first (second) channel to classical (pseudo) bulges, and compare the predicted fractions to observations. We emphasize that our treatment does not include other mechanisms of bulge growth such as intrinsic secular processes in the disc or misaligned gas accretion. Interestingly, we find that the evolution of the stellar and gaseous contents of the satellite as it spirals towards the central galaxy is a key ingredient in setting the morphology of the remnant galaxy, and that a good match to the observed bulge demographics occurs when this evolution proceeds closely to that of the central galaxy.