Strengthening America's Health Care Readiness Act This bill establishes an emergency services demonstration program within the Department of Health and Human Services (HHS) to provide supplemental student loan repayment to current and former National Health Services Corps (NHSC) participants. The NHSC provides scholarships and student loan repayment awards to health care providers who agree to work in areas with health care provider shortages. To receive supplemental payments, individuals must agree to participate in the National Disaster Medical System. This system is a partnership between HHS, the Department of Defense, and other federal departments that responds to public health and other emergencies, including by deploying medical response teams. The bill also increases FY2021 funding for the NHSC, as well as similar programs for nurses. It reserves some of this funding for individuals who are members of groups that are historically underrepresented in health care professions. The funds for the Nurse Corps Scholarship and Repayment Program are designated as an emergency requirement, which exempts the funds from discretionary spending limits and other budget enforcement rules.