This article investigates the leader-follower consensus problem of a class of non-strict-feedback nonlinear multiagent systems with asymmetric time-varying state constraints (ATVSC) and input saturation, and an adaptive neural control scheme is developed. By introducing the distributed sliding-mode estimator, each follower can obtain the estimation of leader's trajectory and track it directly. Then, with the help of time-varying asymmetric barrier Lyapunov function and radial basis function neural networks, the controller is designed based on backstepping technique. Furthermore, the mean-value theorem and Nussbaum function are utilized to address the problems of input saturation and unknown control direction. Moreover, the number of adaptive laws is equal to that of the followers, which reduces the computational complexity. It is proved that the leader-follower consensus tracking control is achieved without violating the ATVSC, and all closed-loop signals are semiglobally uniformly ultimately bounded. Finally, the simulation results are provided to verify the effectiveness of the control scheme.