National Nuclear University Research Infrastructure Reinvestment Act of 2021 This bill reauthorizes through FY2026 and expands the Nuclear Energy University Program within the Department of Energy (DOE) that supports university research and training in nuclear science and engineering. For example, the bill expands the program to provide support to universities for revitalizing and upgrading existing nuclear science and engineering infrastructure that support the development of advanced nuclear technologies and applications. In addition, the bill establishes an Advanced Nuclear Research Infrastructure Enhancement Subprogram under which DOE must establish four new research reactors as well as new nuclear science and engineering facilities.