Delay performance is an important indicator in underwater acoustic sensor networks (UASNs), where both propagation delay and retransmission should be considered. However, the current literatures usually adopt the oversimplified disclike interference model and evaluate the delay performance by system-level simulations. Therefore, in this paper, we propose an analytical model to evaluate the local delay with slotted ALOHA scheduling. Due to the unique 3D characteristics and randomness of the UASNs, the UASNs are modeled as a 3D spatial homogeneous Poisson point process and a more practical interference model in underwater environment is proposed. Based on the statistical characteristics of the signal-interference-plus-noise ratio, we derive a computable representations for the local delay and its upper bound. Simulation results corroborate the tightness of the bound and verify our analysis.