Glynn was the eldest and only surviving son of Robert Glynn of Brodes in Helland parish, near Bodmin, Cornwall, who married Lucy, daughter of John Clobery of Bradstone, Devon, was born at Brodes on 5 August and baptised at Helland Church on 16 September 1719.
Glynn himself announced in March 1751 a course of lectures at King's College on the medical institutes, and next year gave a second course on anatomy.
William Pitt the Younger, whom he had attended in the autumn of 1773, offered him in 1793 the professorial chair of medicine at Cambridge, which Glynn refused.
He was at the close of his life the acknowledged local head of his profession, and his medical services were in great repute at Ely, where he attended every week.
A tablet to his memory was placed in King's College Chapel, in a small oratory on the right hand after entering its south door.
The college received a legacy in stock; it was mainly expended on buildings erected under superintendence of William Wilkins the architect around 1825–30; with a prize of £20 a year divided between two scholars.
George Pryme describes him as usually wearing 'a scarlet cloak and three-cornered hat; he carried a gold-headed cane.
[2] He was friendly with William Mason, and attended Thomas Gray in his last illness; Glynn's initial noncommittal diagnosis of "gout of the stomach" missed terminal uremia.
His kindness to one of his poor patients was celebrated by a younger son of Henry Plumptre, president of Queens' College, in verses called Benevolus and the Magpie.
Horace Walpole called him in 1792 "an old doting physician and Chattertonian at Cambridge," and professed to believe that some falsehoods current about himself had been invented or disseminated by Glynn.
Some stanzas by him beginning "Tease me no more" appeared in the General Evening Post, 23 April 1789, and were reprinted in the Poetical Register for 1802, and Henry John Wale's My Grandfather's Pocket-Book.
[2] Glynn believed in the authenticity of the forged "Rowley poems" published by Thomas Chatterton, and his faith was confirmed by a visit to Bristol in 1778.