This article considers the distributed output-feedback consensus control problem for nonlinear multiagent systems subject to input delays. Different from the existing related works, the input delay of each agent is described as an unknown time-varying function and is different from each other in this article. To deal with this problem, for each follower, we first construct a novel distributed observer based on the relative output information to asymptotically estimate the state information of the leader, then we introduce a classical observer to asymptotically estimate the state information of the follower based on its output information. By means of two observers, the leader-following consensus problem is transformed into the stability problem of the nonlinear system with unknown input delays. Subsequently, the distributed controller independent of delays is proposed for each follower by the use of the truncated prediction method under some conditions. Based on the Lyapunov stability theory, it is strictly proved that the distributed controller can render all agents achieving consensus. Finally, the effectiveness of the theoretical results is illustrated on the basis of numerical simulations on a group of single-link manipulators.