In this paper, we propose an adaptive prescribed-time control algorithm for the fixed-wing unmanned aerial vehicle (UAV). How to follow the desired trajectory within a predetermined time is a problem worth investigating in fixed-wing UAV tracking missions. To this end, a novel method based on time-varying state feedback and segmented neural network (SNN) is proposed, using practice prescribed-time input-to-state stable to guarantee the convergence of all signals in the prescribed time. Considering the input saturation and state constraints, we give the basis for selecting the prescribed time with different initial conditions, rather than an arbitrary one. Finally, the simulation shows that the proposed method can realize prescribed-time tracking control with input saturation, despite large initial states, and the magnitude of the control changes moderately.