National Dam and Hydropower Safety Improvements Act of 2021 This bill requires the Federal Energy Regulatory Commission (FERC) to take certain actions related to dam safety. Specifically, FERC may issue a license for a dam and other project works only if the project meets the relevant safety requirements and the licensee can operate and manage the project works in a manner that ensures dam and public safety. A licensee with an already-issued license must also meet these safety requirements. FERC must also establish procedures to assess the financial viability of an applicant for such a license. Finally, the bill requires FERC's Office of Energy Projects to (1) notify a state in certain situations, such as when FERC issues a finding that requires a dam operator in that state to make repairs; and (2) hold a conference with states to discuss dam safety issues.