In this paper, a lattice Boltzmann model with BGK operator (LBGK) for solving time-fractional nonlinear wave equations in Caputo sense is proposed. First, the Caputo fractional derivative is approximated using the fast evolution algorithm based on the sum-of-exponentials approximation. Then the target equation is transformed into an approximate form, and for which a LBGK model is developed. Through the Chapman-Enskog analysis, the macroscopic equation can be recovered from the present LBGK model. In addition, the proposed model can be extended to solve the time-fractional Klein-Gordon equation and the time-fractional Sine-Gordon equation. Finally, several numerical examples are performed to show the accuracy and efficiency of the present LBGK model. From the numerical results, the present model has a second-order accuracy in space.