Finding equilibria via gradient play in competitive multi-agent games has been attracting a growing amount of attention in recent years, with emphasis on designing efficient strategies where the agents operate in a decentralized and symmetric manner with guaranteed convergence. While significant efforts have been made in understanding zero-sum two-player matrix games, the performance in zero-sum multi-agent games remains inadequately explored, especially in the presence of delayed feedbacks, leaving the scalability and resiliency of gradient play open to questions. In this paper, we make progress by studying asynchronous gradient plays in zero-sum polymatrix games under delayed feedbacks. We first establish that the last iterate of entropy-regularized optimistic multiplicative weight updates (OMWU) method converges linearly to the quantal response equilibrium (QRE), the solution concept under bounded rationality, in the absence of delays. While the linear convergence continues to hold even when the feedbacks are randomly delayed under mild statistical assumptions, it converges at a noticeably slower rate due to a smaller tolerable range of learning rates. Moving beyond, we demonstrate entropy-regularized OMWU -- by adopting two-timescale learning rates in a delay-aware manner -- enjoys faster last-iterate convergence under fixed delays, and continues to converge provably even when the delays are arbitrarily bounded in an average-iterate manner. Our methods also lead to finite-time guarantees to approximate the Nash equilibrium (NE) by moderating the amount of regularization. To the best of our knowledge, this work is the first that aims to understand asynchronous gradient play in zero-sum polymatrix games under a wide range of delay assumptions, highlighting the role of learning rates separation.