Westminster, western part of London has always been the home of peers, bishops and toffs. We can rule out having anything to do with being a ship's carpenter who would probably reside in the eastern part of London. Thus your ancestor must have worked on furniture and fixtures and fittings like this in the City of Westminster:-
"There is a large old house at the end of Cowley Street, having a fine double staircase; indeed, there are fine staircases, and other marks of aristocratic occupation, in many of the houses round about this spot."
https://www.british-history.ac.uk/old-new-london/vol4/pp1-13***
Moving on to whether there is a connection between carpentry in London and foreign lands - I spotted this about one of the most famous antique furniture makers of London - this being "Gillows furniture":-
"1814-1897
... the company expanded from furniture design and bespoke manufacture to being an interior design company that would manufacture furniture and supply other manufacturers products. They were the largest company of its type outside London; where they maintained showrooms and workshops. They had a solid reputation for quality.
The company won commissions to furnish and decorate public buildings in Australia, South Africa, India, Russia, Germany, France and the United States. "
https://en.wikipedia.org/wiki/Gillows_of_Lancaster_and_LondonAfter quickly scanning through that webpage it seems the company could have employed apprentices and craftsmen from several parts of the UK, thus
an open mind when looking for what I consider to be mainly a Welsh surname of Williams (not Williamson as found in the rest of the UK). It's a shortish journey from Wales eastward - or if Williamson family shortened their surname to Williams, they could have travelled down from Lancashire to Westminster, London.
Presumably most documentation would have been destroyed in the London blitz of WWII.
Best Wishes,
Rena