A network of independently trained Gaussian processes (StackedGP) is introduced to obtain predictions of geospatial quantities of interest (model outputs) with quantified uncertainties. The uncertain nature of model outputs is due to model inadequacy, parametric uncertainty, and measurement noise. StackedGP framework supports component-based modeling in environmental science, enhances predictions of quantities of interest through a cascade of intermediate predictions usually addressed by cokriging, and propagates uncertainties through emulated dynamical systems driven by uncertain forcing variables. By using analytical first and second-order moments of a Gaussian process with uncertain inputs using squared exponential and polynomial kernels, approximated expectations of model outputs that require an arbitrary composition of functions can be obtained. The performance of the proposed nonparametric stacked model in model composition and cascading predictions is measured in a wildfire and mineral resource problem using real data, and its application to time-series prediction is demonstrated in a 2D puff advection problem.