Tewksbury was born in West Newbury, Massachusetts, the son of James Gardner Tewksbury and Sarah Jane Whittier. He graduated cum laude from Harvard College (1887) and from Hartford Theological Seminary (1890) and was ordained at the East Somerville, Massachusetts, Congregational Church in 1890. That same year he and his wife, Grace (Holbrook), were appointed by the American Board of Commissioners for Foreign Missions to the North China Mission. Arriving at Tungchou, Tewksbury and his wife served two terms, 1890-1898 and 1899-1906. From 1907 to 1910 he was executive secretary of the Laymen's and Young People's Missionary Movement in New York City.
In 1910 the Tewksbusys returned to China where Elwood served with the World's Sunday School Association as the first secretary for the China Sunday School Union, based in Shanghai. In this role he pioneered several educational strategies. He taught religious pedagogy and Sunday school management in seminaries and institutes throughout the country, introduced mass meetings for training purposes, and advocated athletics in the curriculum of Sunday schools. He developed a graded lesson plan on the model used in the United States, and superintended translation into Chinese of a library of Sunday school books. His wife died in Shanghai in 1938. He died in Philadelphia, Pennsylvania.