Wheel-rail forces are an essential indicator for vehicle safety evaluation. The calculation of wheel-rail forces for the coupled nonlinear train-track-bridge system using the direct numerical integration method is time-consuming, hindering the timely safety and reliability assessment of train operations. In this paper, an efficient method based on the nonlinear autoregressive with exogenous input artificial neural network (NARX-ANN) was proposed to predict the nonlinear wheel-rail force of trains on bridges caused by track irregularity. Firstly, samples of track irregularity time history were randomly generated using the stochastic harmonic function method. The vertical and lateral wheel-rail forces of the coupled nonlinear train-track-bridge system were calculated by numerical integration. Secondly, the NARX-ANN model was established and trained with a small number of randomly selected wheel-rail force samples. Finally, the wheel-rail forces obtained by numerical integration were regarded as the ground truth to verify the prediction accuracy of the NARX-ANN model. The influence of training configuration, initial output, and the number of time delays on prediction accuracy was systematically analyzed. The results showed that the NARX-ANN model could accurately predict the time history of wheel-rail forces. In addition, its computational efficiency is 19.35 times higher than that of the numerical integration method. It is hoped that this study can guide the stochastic analysis of the train–track–bridge system.