This study presents a Sobol-based global sensitivity framework for quantifying uncertainties arising from various sources, including material properties, floor quality and nonlinear connections, numerical definition of pier drift capacity, damping ratio, and record-to-record variability in the seismic assessment of unreinforced masonry buildings (URM). A representative typology of the pre-code building stock in the Lisbon metropolitan area is adopted as a case study. Computational models are developed in OpenSees using the equivalent frame method with three-dimensional macroelements able to capture in-plane (IP) and out-of-plane (OOP) responses of masonry walls, while also incorporating nonlinear constitutive laws for interface elements. Modelling uncertainties are represented through probabilistic distributions of random variables and sampled using Latin Hypercube Sampling (LHS). Hazard-consistent ground-motion records are employed as input for incremental dynamic analyses (IDA) to explicitly account for record-to-record variability. Approximately 30,000 nonlinear dynamic simulations are performed using cloud-based high-performance computing and are further complemented by a HistGradientBoostingRegressor (HGBT) surrogate model to enable statistically robust Sobol sensitivity analyses. The results identify the most influential uncertainty sources governing the variance of key engineering demand parameters (EDPs) across different damage states (DSs). Moreover, the extensive numerical dataset enables a robust characterization of IP and OOP failure modes, the derivation of efficient capacity and fragility functions, and a detailed examination of local failure mechanisms. Overall, the findings emphasize the importance of explicitly accounting for dominant sources of uncertainty in the seismic safety assessment of existing URM buildings.