Recent progress of nucleosynthesis work as well as the discovery of a kilonova associated with the gravitational-wave source GW170817 indicates that neutron star mergers (NSM) can be a site of the r-process. Several studies of galactic chemical evolution, however, have pointed out inconsistencies between this idea and the observed stellar abundance signatures in the Milky Way: (a) the presence of Eu at low (halo) metallicity and (b) the descending trend of Eu/Fe at high (disc) metallicity. In this study, we explore the galactic chemical evolution of the Milky Way's halo, disc and satellite dwarf galaxies. Particular attention is payed to the forms of delay-time distributions for both type Ia supernovae (SN Ia) and NSMs. The Galactic halo is modeled as an ensemble of independently evolving building-block galaxies with different masses. The single building blocks as well as the disc and satellite dwarfs are treated as wellmixed one-zone systems. Our results indicate that the aforementioned inconsistencies can be resolved and thus NSMs can be the unique r-process site in the Milky Way, provided that the delay-time distributions satisfy the following conditions: (i) a long delay (∼ 1 Gyr) for the appearance of the first SN Ia (or a slow early increase of its number) and (ii) an additional early component providing 50% of all NSMs with a delay of ∼ 0.1 Gyr. In our model, r-process-enhanced and r-process-deficient stars in the halo appear to have originated from ultra-faint dwarf-sized and massive building blocks, respectively. Our results also imply that the natal kicks of binary neutron stars have a little impact on the evolution of Eu in the disc.