The son of a merchant, Taswell was born at Cowes. He went up to Christ Church, Oxford on 8 July 1670, gaining his B.A. in 1674, M.A. in 1677, B.D. in 1685 and D.D. in 1698. While there he had a beneficial relationship with the Dean, John Fell, who offered him financial support knowing that he received little financial support from his father. In exchange he assisted Fell in manuscript work, for example translating Thomas Lydiat's Chronological Canons into Latin. In 1681, he was appointed college lecturer in Greek.
