Compared to a single-robot workstation, a multi-robot system offers several advantages: 1) it expands the system's workspace, 2) improves task efficiency, and, more importantly, 3) enables robots to achieve significantly more complex and dexterous tasks, such as cooperative assembly. However, coordinating the tasks and motions of multiple robots is challenging due to issues, e.g., system uncertainty, task efficiency, algorithm scalability, and safety concerns. To address these challenges, this paper studies multi-robot coordination and proposes APEX-MR, an asynchronous planning and execution framework designed to safely and efficiently coordinate multiple robots to achieve cooperative assembly, e.g., LEGO assembly. In particular, APEX-MR provides a systematic approach to post-process multi-robot tasks and motion plans to enable robust asynchronous execution under uncertainty. Experimental results demonstrate that APEX-MR can significantly speed up the execution time of many long-horizon LEGO assembly tasks by 48% compared to sequential planning and 36% compared to synchronous planning on average. To further demonstrate performance, we deploy APEX-MR in a dual-arm system to perform physical LEGO assembly. To our knowledge, this is the first robotic system capable of performing customized LEGO assembly using commercial LEGO bricks. The experimental results demonstrate that the dual-arm system, with APEX-MR, can safely coordinate robot motions, efficiently collaborate, and construct complex LEGO structures. Our project website is available at https://intelligent-control-lab.github.io/APEX-MR/.