This paper is devoted to the approximate consensus problem for stochastic networks of nonlinear agents with switching topology, noisy, and delayed information about agent states. A local voting protocol with nonvanishing (e.g., constant) step size is examined under time-varying environments of agents. To analyze dynamics of the closed-loop system, the so-called method of averaged models is used. It allows us to reduce analysis complexity of the closed-loop stochastic system. We derive the upper bounds for mean square distance between states of the initial stochastic system and its approximate averaged model. These upper bounds are used to obtain conditions for approximate consensus achievement. An application of general theoretical results to the load balancing problem in stochastic dynamic networks with incomplete information about the current states of agents and with changing set of communication links is considered. The conditions to achieve the optimal level of load balancing are established. The performance of the system is evaluated both analytically and by simulation.