This paper investigates the time-dependent sensitivity of structural reliability assessment to multi-source uncertainties using Lamb wave. To precisely model the influence of local damage on the structure in the course of damage growth, a surface damage effect model is proposed to obtain the equivalent elasticity modulus which can be coupled with the structure model. The evolution of the surface damage is modeled using the fatigue crack propagation model. Furthermore, by setting up the component and structure failure criteria, the time-dependent reliability model of the structure under multi-source uncertainties from Lamb wave detection and material properties is established. The method of score function is employed to evaluate the sensitivity index, which is defined as the derivative of the reliability with respect to the distribution parameters of uncertain variables. A spatial truss structure is used to demonstrate the overall procedure. Numerical results show that the sensitivities indices are time- and damage size-dependent. The sensitivity contributions from Lamb wave quantification model and the material properties are comparable when the crack size is relatively small. When the crack grows to a larger size, the reliability assessment result is much more sensitive to uncertainties associated with material properties.