Elder was born in Hexham, England, on June 2, 1947. As a youth, he showed talent both as a singer and instrumentalist, serving as a chorister at Canterbury Cathedral and as the first-chair bassoonist in the National Youth Orchestra. He studied music at Corpus Christi College, Cambridge, and was later mentored by renowned conductor and musicologist Edward Downes. Elder worked briefly in minor capacities at Glyndebourne and Covent Garden before serving as the conductor of Opera Australia from 1972-1974. He returned to England in 1974 to serve as a regular conductor at the English National Opera. In 1979, he was appointed the music director of the ENO; he remained in this position until 1993. During his years as ENO director, Elder conducted the company in numerous highly acclaimed productions and led successful tours to the U.S., Russia, and other parts of Europe. He has also conducted the Chicago Lyric Opera and the Metropolitan Opera, among other important opera companies. Opera Rara, the label committed to forgotten or neglected 19th century operas, named Elder as artistic director in the spring of 2011 to help guide its endeavors; he remained in this role until 2019. Elder has also been active in the orchestral world. He was music director of the Rochester Philharmonic from 1989 until 1994. In 2000, he was appointed music director of the Hallé Orchestra. He served as principal guest conductor for the London Mozart Players (1980-1983), the BBC Symphony Orchestra (1982-1985), and the City of Birmingham Symphony Orchestra (1992-1995). He conducts yearly at the Proms; since 2003, he has conducted the Hallé in these performances. He has also had the honor of conducting the "Last Night of the Proms" on two occasions, both times leading the BBC Symphony. The Orchestra of the Age of Enlightenment named Elder its principle artist in 2011. He has conducted many of the world's leading orchestras, including the Boston, Gothenburg, and Vienna Symphony Orchestras, and the London, Rotterdam, and Bergen Philharmonic Orchestras. His repertory is broad in both the operatic and orchestral realms: though he has favored Verdi in the opera house, he has led performances of Wagner (Die Meistersinger; Parsifal), Busoni (Doktor Faust), Stravinsky (The Rake's Progress), and contemporary composers like David Blake (Toussaint). Elder's orchestral choices include many standards while taking in works by contemporary British composers like George Benjamin, Nicholas Maw, and Jonathan Harvey. He has made numerous recordings for a variety of labels, including Chandos, EMI, and Hyperion. Since 2003, he has also recorded with Hallé on the Hallé label. Among Elder's more successful recordings are his English-language version of Verdi's Rigoletto on Chandos (2000) and Elgar's Symphony No. 2 on the Hallé label (2004). In 2019, he released several albums, including the final opera of his Ring des Nibelungen cycle with Hallé, Wagner's Siegfried, on the Hallé label, and Donizetti's opera L'Ange de Nisida with the Royal Opera House Covent Garden, on Opera Rara.
Elder was awarded the Commander of the Order of the British Empire in 1989, was knighted in 2008, and was appointed a Member of the Order of the Companions of Honour in 2017.