A recently formulated description of homogeneous nucleation for Brownian particles in the over-damped limit based on fluctuating hydrodynamics is used to determine the nucleation pathway, characterized as the most likely path (MLP), for the nucleation of a dense-concentration droplet of globular protein from a dilute solution in a small, finite container. The calculations are performed by directly discretizing the equations for the MLP and it is found that they confirm previous results obtained for infinite systems: the process of homogeneous nucleation begins with a long-wavelength, spatially-extended concentration fluctuation that it condenses to form the pre-critical cluster. This is followed by a classical growth processes. The calculations show that the post-critical growth involves the formation of a depletion zone around the cluster whereas no such depletion is observed in the pre-critical cluster. The approach therefore captures dynamical effects not found in classical Density Functional Theory studies while consistently describing the formation of the pre-critical cluster.