He was born on 19 December 1721 at Barrow-upon-Soar, Leicestershire, where his father (died 1765), later vicar of Sileby and rector of Morton, Derbyshire, was then curate.
After receiving instruction from his father, and studying at Chesterfield grammar school, he entered Jesus College, Cambridge, and graduated B.A.
His publications attracted the notice of William Warburton, who presented Heathcote to the assistant preachership at Lincoln's Inn.
He moved in June 1753 to London, where he associated with John Jortin, Thomas Birch, Matthew Maty, and others, who met once a week to drink coffee and talk learnedly.
When in 1752 he wanted to take a part in the controversy set off by Conyers Middleton on the miraculous powers ascribed to the early Christian Church, he felt a lack of fluency in literary English.
He produced two pamphlets anonymously: Cursory Animadversions on the Controversy in General (1752), and Remarks upon a Charge by Dr. Chapman (1752); and in the following year wrote a reply to Thomas Fothergill's sermon on the uses of commemorating King Charles I's martyrdom.