His father Alexander Thomson was Postmaster-General, a member of the council of the Province of Georgia, and custom-collector for Savannah, Georgia. He was born in Edinburgh where his parents were staying temporarily, went to America but along with other loyalists returned to Britain after the result of the American War of Independence returned to Edinburgh.
In 1828 he became the first professor of materia medica and therapeutics at London University (now University College London). In 1832, on the death of John Gordon Smith, he was appointed joint professor of medical jurisprudence with Andrew Amos. In 1837 Amos was appointed a member of the governor-general's council in India and so Thomson became sole professor.
He married twice. In 1801 he married Christina Maxwell, and they had one son and two daughters, but she died in 1820. In 1820 he married Katherine Byerley, daughter of Thomas Byerley and they had three sons and five daughters. His daughter from this second marriage Elizabeth married Rennell Rodd, 1st Baron Rennell.