For collaborative search missions in confined spaces such as disaster-stricken ruins and industrial pipelines, the formation of small tandem-rotor unmanned aerial vehicles (STR-UAVs) with narrow fuselages and long endurance forms an ideal operational platform. To address the multiple constraints of input saturation, external disturbances, and actuator failures in such complex operational environments, an adaptive distributed prescribed-time fault-tolerant formation control strategy is proposed. Firstly, a novel smooth saturation function is designed to address the issue of input saturation constraints with asymmetric sign. Secondly, a novel prescribed-time sliding mode surface is constructed, achieving the convergence of formation errors within a user-defined time. Thirdly, an adaptive distributed prescribed-time fault-tolerant formation control strategy is proposed by integrating the aforementioned smooth saturation function, prescribed-time sliding mode surface, and adaptive method. Finally, based on Lyapunov theory, the prescribed-time stability of the proposed formation control strategy is rigorously proven, and numerical simulations are conducted. The simulation results not only validate the effectiveness of the proposed control strategy but also highlight its superiority over some existing methods through qualitative analysis and quantitative comparisons.