Cherokee Nation and Chickasaw Nation Criminal Jurisdiction Compacting Act of 2021 This bill authorizes the Cherokee Nation and the Chickasaw Nation to each enter into intergovernmental compacts with Oklahoma for the state to exercise its criminal jurisdiction within Indian country. The bill outlines the requirements for such compacts, including that they must define the categories of criminal offenders or offenses within the reservation over which Oklahoma may exercise its jurisdiction. The Department of Justice must, upon the request of a nation, consult with and provide technical assistance for developing or implementing a compact. A nation may require an offender who is sentenced to a term of imprisonment of more than six months to serve such a sentence in the nearest appropriate federal facility. The Department of the Interior must publish a compact that it receives from a nation. Interior must also report on the development and implementation of compacts and make recommendations to support and enhance the criminal justice system of each nation.