Horace Walpole states that he went in 1725 to Bexley in Kent with his cousins, "the four younger sons of Lord Townshend, and with a tutor, Edward Weston ... and continued there some months."

Weston was secretary to Lord Townshend during the king's residence at Hanover in 1729, and, on his retirement from office, lost "a very generous friend and patron".

In May 1730 he offered his services to Lord Harrington, and when that peer was made secretary of state for the northern department, Weston became under-secretary, remaining in the position until 1746.

In July 1763 he addressed a letter to George Grenville on his ill-health and his sole reward "of £275 per annum, with the honourable title of gazetteer" in the secretary's department.

Weston then served under Lord Halifax in the southern department and recommended the issue of a general warrant against John Wilkes.

He died at Buxton, Derbyshire on 15 July 1770, and was buried at St Margarets' church, Somerby, Lincolnshire, where a monument recorded his memory.