This paper presents a comprehensive method to efficiently solve non-stationary, nonlinear, stochastic integrated assessment models (IAMs) and perform parametric uncertainty quantification. Our approach comprises two main components: a deep learning-based algorithm to solve IAMs globally as a function of endogenous, exogenous, as well as uncertain parameters in a single model evaluation. Secondly, we devise a Gaussian process-based surrogate model to analyze quantities of interest, such as the social cost of carbon, in relation to model input parameters, facilitating streamlined estimation of Sobol' indices, Shapley values, and univariate effects. To demonstrate our method's efficacy, we introduce a high-dimensional stochastic IAM, aligned with cutting-edge climate science. This model incorporates a social planner with recursive preferences, iterative belief updates of equilibrium climate sensitivity via Bayes' rule, and stochastic climate tipping. Uncertainty quantification results reveal a decreasing uncertainty in equilibrium climate sensitivity over time, while the intertemporal elasticity of substitution predominantly influences the social cost of carbon by 2100.