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 recipients of National Health Services Corps (NHSC) scholarships and student loan repayment awards who agree to participate in the National Disaster Medical System. (The NHSC provides scholarships and student loan repayment awards to health care providers who agree to work in areas with health care provider shortages, and the National Disaster Medical 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 FY2020 funding for the NHSC, as well as similar programs for nurses, with a certain amount of funding reserved 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.