Streamtube-based aerodynamic models are widely used for the analysis of vertical-axis wind turbines (VAWT) due to their low computational cost. Among them, the Double Multiple Stream Tube model (DMST) provides an efficient compromise between classical momentum models and high-fidelity CFD simulations. In this work, an analytical extension of the DMST is proposed, addressing a key limitation of standard formulations, which typically neglect the circumferential component of the induced velocity, assuming that the flow can be adequately described using axial (streamwise) components only through the rotor disks. This work improves the DMST formulation to explicitly account for the lateral velocity component within the multiple streamtubes associated with each disk. The proposed formulation introduces a lateral induction factor that captures the previously neglected cross-flow interactions and their effect on blade aerodynamics, dependent on the rotor azimuthal position and the local streamtube discretization, allowing a more accurate evaluation of the effective angle of attack and aerodynamic loads on the blades. The extended model preserves the algebraic structure and low computational cost of the classical DMST and can be solved using standard iterative schemes. Comparisons with two-dimensional CFD results of the full rotating turbine show that the inclusion of circumferential velocity effects improves the prediction of torque and power, with especial enhancement on the estimation of the unsteady angle-of-attack on the turbine blades, particularly for off-design operating conditions and highly discretized rotors. Finally, the DMST results are extrapolated to a newly proposed turbine prototype with a 2 m diameter, employing the same blade airfoil and solidity, as a proof of applicability for preliminary design purposes. The proposed model enhances the applicability of DMST for parametric studies and preliminary VAWT design.