Immigrants in Nursing and Allied Health Act of 2023 This bill authorizes and expands programs to reduce barriers to enter the nursing or allied health professional workforce that are faced by some immigrants. Specifically, the Department of Health and Human Services may award grants to state, tribal, and local governments and private organizations for covering certain costs to assist lawfully present immigrants with entering such professions. These include costs related to education, training, or licensure and assistance in obtaining overseas academic or training records. The bill also makes lawfully present immigrants eligible to participate in the National Health Services Corps (NHSC). The NHSC provides scholarships and student loan repayment awards to health care providers who agree to work in areas with health care provider shortages.