During the vitrification of nuclear waste enriched in molybdenum, phase separation may occur following the nucleation and growth regime, with the molybdenum segregating to the daughter phase. The present work, as part of the CEA project SIVIT, aims to model and simulate this phase separation at the scale of the interface separating the two liquid phases of the melted glass. To this end, the modelling must satisfy a number of constraints: tracking a fully-resolved interface; accounting for the chemical diffusion (which drives the growth dynamics) and the flow dynamics and the effects of each on the motion of the interfaces; and the respect of the equation of state of the glass (here, the model ternary glass Na2O-SiO2-MoO3). Simulations of the model must also have a high numerical efficiency to allow for comparisons in three dimensions and at a satisfying scale with experimental observations. To fulfil these requirements, we formulate a model based on the phase-field theory with a grand potential formulation coupled to the diffusion of the chemical components and to the incompressible Navier-Stokes equations. We discretize this model with the lattice Boltzmann method and implement it in a new high-performance simulation code, LBM_saclay, able to exploit the multi-GPU architectures of modern supercomputers. We then demonstrate the capability of the model to quantitatively reproduce the growth dynamics after nucleation and the influence of flow and sedimentation on these dynamics.