Of unknown parentage, Thomson was born and educated in Wolverhampton.
He came to the notice of John Russell, 4th Duke of Bedford, who appointed him auditor of his estates.
A close friend of Lord Chancellor Thurlow, Thomson was appointed a Master in Chancery in 1782, Accountant-General in 1786, and a Baron of the Exchequer in 1787; he was made a serjeant and knighted the same year.
In 1814, Thomson succeeded Sir Vicary Gibbs as Chief Baron of the Exchequer, and was sworn of the Privy Council.
Unmarried and having been predeceased by his sister in whose favour his will was made, his death led to a lawsuit in chancery.