Automating large-scale manufacturing in domains like timber construction requires multi-robot systems to manage tightly coupled spatiotemporal constraints, such as collision avoidance and process-driven deadlines. This paper introduces LASER (Level-based Asynchronous Scheduling and Execution Regime), a complete framework for scheduling and executing complex assembly tasks, demonstrated on a screw-press gluing application for timber slab manufacturing. Our central contribution is to integrate a barrier-based mechanism into a constraint programming (CP) scheduling formulation that partitions tasks into spatiotemporally disjoint sets, which we define as levels. This structure enables robots to execute tasks in parallel and asynchronously within a level, synchronizing only at level barriers, which guarantees collision-free operation by construction and provides robustness to timing uncertainties. To solve this formulation for large problems, we propose two specialized algorithms: an iterative temporal-relaxation approach for heterogeneous task sequences and a bi-level decomposition for homogeneous tasks that balances workload. We validate the LASER framework by fabricating a full-scale 2.4m x 6m timber slab with a two-robot system mounted on parallel linear tracks, successfully coordinating 108 subroutines and 352 screws under tight adhesive time windows. Computational studies show our method scales steadily with size compared to a monolithic approach.