This paper reports on results concerning the global robust stability and global asymptotic stability of switched nonlinear positive time-varying delay systems when all subsystems are unstable. Based on the assumptions of admissible sector nonlinearity and interval uncertainty, a new nonlinear multiple discretized copositive Lyapunov-Krasovskii functional was constructed, and a mode-dependent dwell time switching technique was utilized to derive sufficient conditions for global robust stability of the system under consideration. Furthermore, when the considered systems did not have without interval uncertainty, adequacy criteria were also proposed to ensure global asymptotic stability. By comparing with previous results, it was verified that our results were less conservative and more general than those of older studies. The theoretical results obtained were presented in illustrative examples applicable to delayed neural networks.