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In this paper, we present an effective online tube-based model predictive control (T-MPC) solution for autonomous driving that aims at improving the computational load while ensuring robust stability and performance in fast and disturbed scenarios. We focus on reformulating the non-linear original problem into a pseudo-linear problem by transforming the non-linear vehicle equations to be expressed in a Linear Parameter Varying (LPV) form. An scheme composed by a nominal controller and a corrective local controller is propossed. First, the local controller is designed as a polytopic LPV-H$_{\infty}$ controller able to reject external disturbances. Moreover, a finite number of accurate reachable sets, also called tube, are computed online using zonotopes taking into account the system dynamics, the local controller and the diturbance-uncertainty bounds considered. Second, the nominal controller is designed as an MPC where the LPV vehicle model is used to speed up the computational time while keeping accurate vehicle representation. Employing reachability theory with zonotopes, the MPC changes online its state and input constraints to ensure robust feasibility and stability under exhogenous disturbances. Finally, we test the presented scheme and compare the local controller performance against the LQR design as state of the art approach. We demonstrate its effectiveness in a disturbed fast driving scenario being able to reject strong exogenous disturbances and fulfilling imposed constraints at a very reduced computational cost.