This paper investigates the prescribed-time coordination problem for multiple Lagrangian systems (MLSs) in the presence of input-to-state redundancy, uncertain dynamic terms, and external disturbances. Moreover, the interdependencies among multiple Lagrangian plants with multi-dimensional states are characterized by matrix-weighted networks. Then, a novel hierarchical prescribed-time control (HPTC) algorithm comprising two parts as well as a hierarchical control framework consisting of two layers are provided to address the aforementioned problem. By virtue of the Lyapunov stability and the nearest neighbor-interaction rules, several sufficient conditions for achieving prescribed-time coordination of the MLSs with input-to-state redundancy and matrix-weighted networks are obtained. Additionally, the designed HPTC algorithm is extended to the case of directed matrix-weighted communication networks in the presence of a directed spanning tree with the leader as the root. Eventually, numerical simulations are provided to demonstrate the validity of the obtained theoretical results.