The paper trail is short and it runs backwards from the present, not forwards from antiquity. Elisha Bartlett’s Essay on the Philosophy of Medical Science (1844) credits “the golden axiom of Chomel” — the Parisian clinician Auguste François Chomel — with the idea that doing good is only the second law of therapeutics, the first being not to do harm. Worthington Hooker repeated the credit in Physician and Patient (1847), but, as Wikipedia’s account of the phrase notes ↗, Hooker “used neither the specific expression, nor the traditional Latin phrase.”
Sixteen years after Bartlett, Thomas Inman’s Foundation for a New Theory and Practice of Medicine (1860) put the English maxim and a Latin aphorism together and hung them on Thomas Sydenham, dead since 1689. Cedric M. Smith, searching writings back to the Middle Ages, found nothing earlier — which makes 1860 Smith’s date for the English-plus-Latin pairing, not for the Latin itself. Suss’s Box 17 prints “Primum non nocere” in medical books from Boyer in 1816 onward, and records that Inman’s own Latin was the looser “Primum est ut non nocere.” Suss also checked the Sydenham and Chomel attributions for primary evidence and reported that neither gives any written source from those physicians or their followers. What is left is a proverb: no author, many wordings, and a borrowed pedigree.