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Grimwade suggests a connection between William Dawson (page 486) and Elizabeth Dawson (page 293) on the circumstantial evidence of the name and the fact that William, a watch maker, died in 1778 and Elizabeth was his widow. No factual evidence has so far come to light to support the proposal.
He entered marks from 1781-1819 at Goldsmiths’ Hall as a case maker from Gutter Lane.
Page 77 of Watchmakers and Clockmakers of the World lists William Dawson as a freeman of the Clockmakers Company from 1778, a liveryman from 1801 and deceased in 1820.
The Will of William Dawson, late watch case maker of King’s Head Court, Gutter Lane, was granted probate on 28th January 1820 (National Archives reference PROB11/1624/417).

Perhaps a need for a William Dawson(I) and a William Dawson(II)?

Committee, 19th March 1778

The Clerk informs the Committee that, in pursuance of information received at the Assay Office that William Dawson, of Oxford Arms Passage, Warwick Lane, watch-case maker, had used forged and counterfeited stamps, in stamping and marking watch-cases, resembling the stamps used by this Company, he has by virtue of a search warrant searched the house of the said William Dawson, and in his presence found such counterfeit stamps with a watch-case stamped therewith; and that he has caused him to be committed to Wood Street Compter, to take his trial for such offence ; and he requests the further direction of the Committee. Whereupon a motion is made, and question put, “ That the Clerk of this Company do at the next session of Oyer and Terminer to be holden for the City of London prosecute the said William Dawson for having in his custody such counterfeit stamps, and also for stamping and marking a watch-case with such stamps against the form of the Statute, and place the charges of such prosecution to the Assay Office account ” ; and the same is resolved in the affirmative unanimously, and ordered accordingly.

Source: Memorials of the Goldsmiths' Company; Being Gleanings from their Records Between the Years 1335 and 1815 - by Walter Sherburne Prideaux - 1896

See also: https://www.oldbaileyonline.org/browse. ... 7780603-66 for details of the trial.

Likely Elizabeth's entry of her mark just a few days after the trial was to ensure the continuation of the business, and not as Grimwade suggested, that she was now a widow.

I would have thought it unlikely that a person found guilty of Forgery and receiving a five-year sentence would later be elected to the Livery of Clockmakers Company, but.... Perhaps Grimwade was correct that William Dawson of Gutter Lane was a son of William(I).