<p>Recent advances in research have made global sensitivity analysis of very large and highly linear life cycle assessment systems feasible. In this paper, we build on these developments to include sensitivity analysis of correlated parameters and nonlinear models. We augment numerical uncertainty propagation with Monte Carlo simulations (i) to include propagation of uncertainty from uncertain variables in parameterized inventory datasets; (ii) to account for correlations between process inputs and outputs and in particular incorporate the carbon balance of combustion activities; (iii) to employ published time-series data instead of static values for electricity generation market mixes in Europe; (iv) to ensure that inputs which are supposed to reach a fixed total (e.g., the percentage contributions of power sources to an electricity mix) actually do so consistently by using the Dirichlet distribution. We then iterate on existing global sensitivity analysis protocols for high-dimensional systems to improve their computational performance. To correctly calculate sensitivity rankings for correlated inputs, we use SHapley Additive exPlanations as feature importance metrics with gradient boosted trees. Our results for a case study of climate change impacts of an average Swiss household confirm that neglecting correlations limits the validity of uncertainty and sensitivity analysis. Our methodology and correlated sampling modules are given as open source&#xa0;code.</p>