This work is the second part of a project that attempts to analyze the formation of Solar system analogues (SSAs) from the gaseous to the post-gas phase, in a self-consistently way. In the first paper (PI) we presented our model of planet formation during the gaseous phase which provided us with embryo distributions, planetesimal surface density, eccentricity and inclination profiles of SSAs, considering different planetesimal sizes and type I migration rates at the time the gas dissipates. In this second work we focus on the late accretion stage of SSAs using the results obtained in PI as initial conditions to carry out N-body simulations. One of our interests is to analyze the formation of rocky planets and their final water contents within the habitable zone. Our results show that the formation of potentially habitable planets (PHPs) seems to be a common process in this kind of scenarios. However, the efficiency in forming PHPs is directly related to the size of the planetesimals. The smaller the planetesimals, the greater the efficiency in forming PHPs. We also analyze the sensitivity of our results to scenarios with type I migration rates and gap-opening giants, finding that both phenomena act in a similar way. These effects seem to favor the formation of PHPs for small planetesimal scenarios and to be detrimental for scenarios formed from big planetesimals. Finally, another interesting result is that the formation of water-rich PHPs seems to be more common than the formation of dry PHPs.