Protecting the Health and Wellness of Babies and Pregnant Women in Custody Act This bill establishes requirements to address the health needs of incarcerated women related to pregnancy and childbirth. The bill requires the Bureau of Justice Statistics to collect data on the health needs of incarcerated pregnant women at the federal, state, tribal, and local levels. With respect to incarcerated women at the federal level, the bill requires the Bureau of Prisons (BOP) to provide appropriate services and programs to address the health and safety needs related to pregnancy and childbirth, as well as appropriate health care to a woman with a high-risk pregnancy; limits the use of restrictive housing for prisoners who are pregnant or in postpartum recovery; prohibits the use of solitary confinement for an incarcerated pregnant woman in her third trimester; requires the BOP to annually report on the number of administrative claims and appeals filed by pregnant inmates; and requires the Government Accountability Office (GAO) to study the services and protections for pregnant incarcerated women in federal pretrial detention facilities. With respect to incarcerated women at the state and local level, the bill requires the GAO to study the services and protections for pregnant incarcerated women in state and local correctional settings.