In 1734 he succeeded Obadiah Shuttleworth as organist of St Michael, Cornhill in London; in 1736 he resigned this post to succeed John Weldon as organist of St Martin-in-the-Fields, London. Charles Burney wrote that Handel was among the musicians who visited St Martin's to hear him play. In 1739 Kelway became a founding governor of the Royal Society of Musicians.
His pupils included Charles Wesley and Mrs Mary Delaney, and he was appointed harpsichord master to Queen Charlotte, wife of George III, at the time of her arrival in England in 1761. Richard, 7th Viscount FitzWilliam was a patron.
As a harpsichordist, Burney wrote that Kelway was "head of the Scarlatti sect". His only notable publication was Six Sonatas for Harpsichord (1764), influenced by Scarlatti. Towards the end of his life, his music went out of fashion: a new style was introduced by Johann Christian Bach, and the pianoforte became fashionable.
- "Kelway [Kellaway, Kellway], Joseph". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/15310.(Subscription or UK public library membership required.)
- Middleton, Louisa M. (1892). Lee, Sidney (ed.). Dictionary of National Biography. 30. London: Smith, Elder & Co. pp. 360–361. . In
- Middleton, Louisa M. (1897). Lee, Sidney (ed.). Dictionary of National Biography. 52. London: Smith, Elder & Co. p. 175. This article states that Shuttleworth retained the post until his death (1734). The 1892 DNB article for Kelway states "about 1730"; this date is repeated in the online DNB article. . In