The numerical solution of risk-averse optimization problems constrained by PDEs requires substantial computational effort resulting from the discretization of the underlying PDE in both the physical and stochastic dimensions. To practically solve these challenging optimization problems, one must intelligently manage the individual discretization fidelities throughout the optimization iteration. In this work, we combine an inexact trust-region algorithm with the recently developed local reduced-basis approximation to efficiently solve risk-averse optimization problems with PDE constraints. The main contribution of this work is a numerical framework for systematically constructing surrogate models for the trust-region subproblem and the objective function using local reduced-basis approximations. We demonstrate the effectiveness of our approach through several numerical examples.