We construct an I-function for toric bundles obtained as a fiberwise GIT quotient of a (not necessarily split) vector bundle. This is a generalization of Brown's I-function for split toric bundles and the I-function for non-split projective bundles. In order to prove the mirror theorem, we establish a characterization of points on the Givental Lagrangian cones of toric bundles and prove a mirror theorem for the twisted Gromov-Witten theory of a fiber product of projective bundles. The former result generalizes Brown's characterization for split toric bundles to the non-split case.