It is commonly believed that uncertainties are obstacles to the tracking performances of flexible air-breathing hypersonic vehicles (FAHVs). In addition, the estimation of unmeasured states in an FAHV makes this issue much more complicated. To deal with these difficulties, this article explores a novel adaptive fuzzy nonsmooth backstepping output-feedback control scheme for the FAHV with closed-loop finite-time convergence. First, to approximate the unknown dynamics of the FAHV, some function approximators are constructed by utilizing an interval type-2 (IT2) fuzzy logic system. On this basis, a fixed-time convergent adaptive IT2 fuzzy observer is designed to estimate the unmeasured flight path angle and the unmeasured angle of attack of the FAHV accurately and rapidly. Consequently, the estimation errors of the designed observer are convergent in a fixed time independent of their initial estimation errors. Furthermore, based on the estimated states, velocity and altitude tracking controllers are designed by using a finite-time adaptive IT2 fuzzy nonsmooth backstepping control technique, which avoids the problem of "explosion of complexity" in traditional backstepping methods. Subsequently, rigorous Lyapunov stability analysis is conducted to show the finite-time convergence of the closed-loop signals of the FAHV control system. Finally, the robustness and superiority of the proposed control scheme is validated by several simulations in representative scenarios.