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This paper deals with finite-time formation control problems of unmanned aerial vehicle swarm system with time-delay and input saturation. Using precise feedback linearization, first, the nonlinear time-delay model of unmanned aerial vehicle is transformed into a linear second-order time-delay system. Based on the neighbors’ states, a fixed-time distributed observer is constructed to estimate the leader’s state for each follower unmanned aerial vehicle quickly and accurately. Moreover, by utilizing Artstein’s transformation, the delayed second-order system is transformed into a delay-free system and a saturation function is used to tackle the input saturation problem. Then, a finite-time formation control protocol is designed based on the Artstein’s transformation and saturation function. Rigorous proof shows that all control inputs are bounded and the formation control with time-delay is achieved in finite time. Finally, a simulation example is given to verify the effectiveness of the proposed protocol.