The crack tip mechanical properties of welded joints are essential factors that influence the failure and stress corrosion cracking (SCC) behaviors of nuclear structures. Owing to the inherent statistical characteristics of the load, material properties, and geometric parameters, the mechanical properties of the crack tips of the base and weld metals in welded joints are uncertain and inhomogeneous. Because welded joints are prone to failure and SCC, a probabilistic methodology for determining the crack tip mechanical properties based on the elastic–plastic finite element and sparse polynomial chaos expansion methods was developed to investigate the influence of material mismatch and uncertainties in welded joints on the failure probability and SCC growth rate. To process the results more efficiently, MATLAB was used for pre- and post-processing during the automated simulation procedure using ABAQUS finite element software. The mechanical properties of the welded joints, such as their J-integrals and plastic strain zones, were examined. Additionally, the failure probabilities and SCC growth rates were determined based on the statistical characteristics of the mechanical properties of the welded joint. The material mismatch and uncertainties in a welded joint have a significant influence on the probabilistic distribution of the mechanical properties of the crack tip, thereby influencing its failure probability and SCC growth rate.