Wright was born near Ludlow in Tenbury Wells, Worcestershire, to a Quaker family who had formerly lived in Bradford. He attended Ludlow Grammar School before graduating from Trinity College, Cambridge, in 1834. While in Cambridge, he contributed to the Gentleman's Magazine and other periodicals, and in 1835 he moved to London to pursue a literary career. In 1842, he was chosen corresponding member of Paris' Academie des Inscriptions et Belles Lettres, as well as a fellow of the Society of Antiquaries and a member of numerous other distinguished British and international organizations. In 1859, he oversaw the excavations of the Roman town of Viroconium Cornoviorum (Wroxeter), near Shrewsbury, and produced a report. The Drawing Room Portrait Gallery contains a portrait of him from October 1, 1859. Thomas Edward Bridgett, an English clergyman and historian, commented, "It is only when he has to speak of the Catholic Church that he is bitter and unfair."
