In this work we consider nucleation in existing and novel lattice models of particles in solution, reviewing, testing and applying the modern methodology for free energy calculation and calculation of nucleation rates based on elements of classical nucleation, reaction rate and transition path theories. We introduce a multi-component lattice model where, at low temperatures, the solute phase can form three distinct solid structures, for which we accurately map the phase diagram, discussing the relevance of the model to the study of nucleation of polymorphic minerals. By analysing multi-dimensional free energy profiles, computed via a path sampling based Monte Carlo protocol, we demonstrate that solute precipitation in the developed model can proceed via nonclassical pathways, where the formation of nuclei of unstable solute phases is followed by their transformation into the thermodynamically preferred structure. Despite the existence of nonclassical nucleation pathways, we show that the conceptual framework of classical nucleation theory provides an adequate quantitative treatment of the nucleation process in our model over a broad range of parameter space.