Eliot, the son of a Writer to the signet, was born in Edinburgh in 1736, and, after education under Nathaniel Jesse, became assistant to a London apothecary.
A fellow Scot, Sir William Duncan, then the King's Physician, gave him help, and he soon made a large income.
In the 1760s Elizabeth Ogborne was born to a tea dealer in London and she reported that Sir John Eliot was her father.
He died, 7 November 1786, at Brocket Hall, Hertfordshire, the seat of his friend Peniston Lamb, 1st Viscount Melbourne.
He was buried in the parish church of Bishops Hatfield, and a tablet to his memory, with some lines by Edward Jerningham on it, was put up by his uncle, William Davidson.