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He signed Indentures as William Strowd in 1769 to be the apprentice of John Wirgman.
In 1782 he was referred to as William Strowd Citizen and Goldsmith, silver polisher of Gutter Lane on the Apprentice Indentures of Thomas Rowe.
In 1783 he signed the Apprentice Indentures of James son of Richard Charnock, goldsmith of Little Britain, London as William Strowd of Gutter Lane.
In 1799 and 1802 he was referred to as William Stroud, Citizen and Goldsmith, silversmith and polisher of Burleigh Street in the Apprentice Indentures signed by William Lewis and Charles Lee respectively.
In 1806 the Indentures signed by William Mountain referred to him as William Strowd, silversmith of Burleigh Street and in the same year he again signed Strowd on the Indentures of Charles Pezzey.
Grimwade’s conjecture that William Strowd, goldsmith of Burleigh Street 1792-1804 was connected with the William Stroud who entered his first mark at Goldmiths Hall from Burleigh Street in 1788 is therefore correct as they appear to be one and the same person.
The burial in 1820 aged 64 years of a William Stroud late of Boyds Gardens was recorded at St George, Hanover Square and may be relevant.

Image of the indenture tying Charles Lee to William Stroud:

A 'J. Strowd' is recorded as a Working Gold and Silversmith of 4, Burleigh Street, Strand.

There were only twenty houses in Burleigh Street at this time, and of which only four appear to be commercial properties, so this is very likely to be William Stroud's business in continuation by another member of the family following the probable retirement of William.

Source: Johnstone's London Commercial Guide - Corrected to 31st August 1817 - Andrew Johnstone - 1818

The 'J. Strowd' mentioned above would be John Strowd, recorded as a Working Silversmith of 4, Burleigh Street, Strand.

Source: Kent's Original London Directory - 1823