A robust fast method for mobile-immobile variable-order (VO) time-fractional diffusion equations (tFDEs) is developed, superiorly handling the cases of small or vanishing lower bound of the VO function. The fast finite difference scheme sharp decreases the storage requirement and computational complexity. Significantly, the error estimate for the proposed scheme is studied without any regularity assumptions of the true solution. Numerical experiments are tested to verify the effectiveness of the proposed method.