Meanwhile, he had begun singing with the Monteverdi Choir, the Schutz Choir, Cantores in Ecclesia, the John Alldis Choir, the Baccholian Singers of London, the Deller Consort, and the Consort of Musick. In 1974, he became a founding member of the Hilliard Ensemble and became more associated with early music performance. He joined the Early Music Consort of London and the London Early Music Group, of which he was also a founding member in 1976. He also recorded and gave concerts with the Academy of Ancient Music, Pro Cantione Antiqua, Broadside Band, and the Medieval Ensemble of London. From 1972, he began taking solo parts in performances of the various ensembles with which he sang and sang extensively in oratorios and at concert engagements throughout Europe. He sang the role of Abaris in Rameau's 1978 Les Boreades in the Flanders Festival and thereafter embarked on a career singing Baroque and Classical-era operas. He made his United States solo debut singing in Handel's Messiah at the Hollywood Bowl, with Christopher Hogwood conducting the Los Angeles Philharmonic Orchestra.
He decided to resettle in the United States, and in 1984 and 1985, he was artist-in-residence at Washington University, St. Louis, MO. The next year, he served as visiting lecturer at Indiana University in Bloomington, IN. In 1986, he became an associate professor of music at the Early Music Institute of Indiana University. Since 1991, he has been a full professor of music at the Early Music Institute and professor of voice.
In 1991, he joined Theatre of Voices, a vocal ensemble founded by Paul Hillier, with whom he had worked in the Hilliard Ensemble. Hillier was then located in Northern California, but in 1996 relocated himself and Theatre of Voices to Bloomington, where Hillier became director of the Early Music Institute. Since Theatre of Voices concerns itself both with early music and recent new music, Elliott became more involved with contemporary music in the last decade of the twentieth century. He appeared as a soloist in the Nonesuch Records release of Steve Reich's Proverbs and in music of John Cage, Arvo Pärt, and Renaissance composers William Byrd, Orlande de Lassus, Thomas Tallis, and Josquin Desprez. His more than 90 recordings appear on the Nonesuch, Harmonia Mundi, and ECM record labels. He makes frequent appearances at festivals and with orchestras in Europe and Britain, and teaches at master classes and workshops throughout North America. He lives in Bloomington with his wife, viola da gamba player Wendy Gillespie, and their two young sons. He is also a licensed real estate broker and bills himself as "the Singing Realtor."