Lawson entered the Royal Military Academy, Woolwich, on 17 July 1758, and passed out as a lieutenant-fireworker, royal artillery, on 25 December 1759. Lawson served through the famous siege of Belle Isle in 1761, and was afterwards at Gibraltar for some years. In 1766, he was promoted to second lieutenant, and in 1771 to first lieutenant. He went to America with Lord Cornwallis in 1776 and was deputy-bridgemaster of the army under Sir William Howe, and in 1779 was appointed bridgemaster to Sir Henry Clinton. There is little information respecting his services during the American war, but in the Royal Military Repository, Woolwich, was placed a model of "a field-carriage for small mortars to be used occasionally as howitzers", which is stated to have been invented and used by him at the siege of Charleston, and another showing his plan of mounting mortars for firing at various elevations, "experimented and approved at New York in 1780". During his time in America he was promoted to captain-lieutenant in 1779, and captain in 1782.