In this article, an adaptive tracking control scheme of deep-stall recovery is proposed for the fixed-wing unmanned aerial vehicle (UAV) with external disturbances and system uncertainties. By analyzing the phase plane of the longitudinal static instability UAV system, the causes and the recovery conditions of the deep-stall are given. Considering the system uncertainties, the external disturbances, and input saturation, a practical prescribed-time (PPT) deep-stall recovery controller is designed by using the time-varying multi-constraints, which contain three layers: Prescribed performance functions (PPFs), actual constraints, and virtual constraints. The new PPT saturation disturbance observer (SDO) and neural networks are employed to deal with the external disturbances and the system uncertainties, respectively. Moreover, the problem of "explosion of complexity" and input saturation is solved by introducing a command filter and an auxiliary system. Then, a new PPT time-varying barrier Lyapunov (PPT-TVBLF) is presented to guarantee the closed-loop system stability under the deep-stall recovery process. Furthermore, simulation study results are given to illustrate the validity of the proposed deep-stall recovery control scheme.