Son of Peter Wedderburn (1700-1756), Lord Chesterhall, and Janet Ogilvy. Married (1767) to Elizabeth Anne Dawson (d. 1781) and (1782) to Charlotte Courtenay.
Wedderburn studied law at the University of Edinburgh and was admitted a member of the Inner Temple, London, in 1753. He became an elder in the General Assembly of the Kirk of Scotland at the young age of 21, and at about the same time he co-founded the Select Society as its youngest member. Member of Parliament, elected for different boroughs, from 1761 and onwards (Ayr Burghs from 1761 to 1768). Solicitor general from 1771 to 1778. Attorney General from 1778. Lord High Chancellor from 1793. Created Lord Loughborough in 1780. Created Earl of Rosslyn in 1801.