Joe Thomas Nelson, physician, was born on October 14, 1923, in Munday, Texas, the son of Walter Clyde and Blanche N. (Tompkins) Nelson. He attended Knox County schools and served in the United States Navy (1942–46). He earned a bachelor's degree from North Texas State University (now the University of North Texas) in 1948 and a medical degree from Baylor University College of Medicine in 1951. After an internship at John Peter Smith Hospital in Fort Worth, Nelson served the community of Weatherford as a family physician for twenty-five years.
He was a member of the Parker County Medical Society, the Texas Medical Association, the Texas Academy of Family Physicians, the Southern Medical Association, the American Academy of Family Physicians, and the American Medical Association. He served as chairman of the Texas Medical Association Council on Legislation (1965–70), a member of the TMA Executive Board (1962–77), and a member of the TMA Board of Trustees (1972–77). In recognition of this distinguished service, the TMA awarded Nelson its Distinguished Service Award in 1976. Fellow members of the Southern Medical Association awarded Nelson their Distinguished Service Award in 1970 and elected him their president for 1973–74. He also served as a member of the AMA Board of Trustees (1974–77).
Nelson provided extraordinary service to educational institutions and community organizations in Texas. He was a member of the North Texas State University Board of Regents (1959–70) and the University of Texas System Board of Regents (1971–77). He received the Outstanding Alumnus award from Baylor College of Medicine in 1973 and the President's Outstanding Alumnus award from North Texas State University in 1976.
In Weatherford Nelson was chief of staff at Campbell Memorial Hospital. He also served nine years on the city's planning and zoning board, eighteen years as a director of Citizen's National Bank, and thirteen years as a director of the Mutual Building and Loan Association. Weatherford named Nelson its Outstanding Citizen in 1963. He was a member of the West Texas Chamber of Commerce and All Saints Episcopal Church. In 1947 Nelson married Varina LeBeau, and they were the parents of a daughter. Nelson died in Dallas on April 11, 1977.