Fueling Our Nuclear Future Act of 2022 This bill modifies provisions under the Energy Act of 2020 concerning high-assay, low-enriched uranium (HALEU), including by directing the Department of Energy (DOE) to (1) establish a program to accelerate the availability of commercially produced HALEU in the United States, and (2) accelerate or initiate activities to make HALEU available from DOE's inventories for use by advanced nuclear reactors.