This paper solves the finite-time switching control issue for the nonstrict-feedback nonlinear switched systems. The controlled plants contain immeasurable states, arbitrarily switchings, and the unknown functions which are constructed with the whole states. Neural network is used to simulate the uncertain systems and a filter-based state observer is designed to estimate the immeasurable states in this paper, respectively. Based on the backstepping recursive technique and the common Lyapunov function method, a finite-time switching control method is presented. Due to the developed finite-time control strategy, the closed-loop signals can be ensured to be bounded under arbitrarily switchings, and the outputs of systems can quickly track the desired reference signals in finite time. The effectiveness of the proposed method is given through its application to a mass-spring-damper system.