When propagating uncertainty in the data of differential equations, the probability laws describing the uncertainty are typically themselves subject to uncertainty. We present a sensitivity analysis of uncertainty propagation for differential equations with random inputs to perturbations of the input measures. We focus on the elliptic diffusion equation with random coefficient and source term, for which the probability measure of the solution random field is shown to be Lipschitz-continuous in both total variation and Wasserstein distance. The result generalizes to the solution map of any differential equation with locally Hölder dependence on input parameters. In addition, these results extend to Lipschitz continuous quantities of interest of the solution as well as to coherent risk functionals of these applied to evaluate the impact of their uncertainty. Our analysis is based on the sensitivity of risk functionals and pushforward measures for locally Hölder mappings with respect to the Wasserstein distance of perturbed input distributions. The established results are applied, in particular, to the case of lognormal diffusion and the truncation of series representations of input random fields.