Here - https://en.wikipedia.org/wiki/Guilloché#In_today’s_terminolo...
it is said that in watchmaking the term for what in English is called 'GUILLOCHÉ WORK' is 'DAMASKEENING', the article on which - https://en.wikipedia.org/wiki/Damaskeening - says
"In horology, DAMASKEENING IS DECORATIVE PATTERNING ON A WATCH MOVEMENT. The term damaskeening is used in America, while in Europe the terms used are Fausses Côtes, Côtes de Genève or Geneva Stripes. Such patterns are made from very fine scratches made by rose engine lathe using small disks, polishing wheels or ivory laps. These patterns look similar to the results of a spirograph or Guilloché engraving."

Note the comment about US vs Europe. "Geneva stripes are also known as “Côtes de Genève” or even “Geneva waves”, and are perhaps among the best known of all the movement decorations. You’ll find them on various movement plates as well as on rotors"
https://www.fratellowatches.com/how-watches-work-what-are-ge...

For European/British term for the profession of creating Geneva stripes, I don't know... Hence medium confidence only.

The 13th census of the USA lists "DAMASKEENER, watch factory" as a profession:
https://books.google.fr/books?id=Z2nSnObtJ9wC&pg=PA190&lpg=P...

However, I cannot swear that that is not the equivalent of 'démasquineur', which is different:
"Le damasquinage (de damaschino, mot italien dérivé de Damas) est une technique de décoration, qui consiste à ENCHÂSSER UN FIL DE CUIVRE, D'OR OU D'ARGENT, sur une surface métallique, généralement en fer ou en acier, et plus anciennement en bronze, afin de créer différents motifs décoratifs et ornementaux. Le résultat de cette incrustation est appelé une damasquinure et l'artisan réalisant ces incrustations est un damasquineur."

A 'sellier' works with leather, but does rather finer work in watchmaking than a saddler!

"I'm searching for a top lEATHERWORKER to complete contract work (leather bands) FOR A WATCHMAKER in Connecticut making small numbers of handmade wristwatches"
"The WATCH IS MOUNTED ON A LEATHER CUFF STRAP handmade by Shrewsbury-based LEATHERWORKER Christopher Clarke"

"Visit any HIGH-END WATCH brand’s facilities and you will discover how the Swiss love to be environmentally friendly: [...] but ask them about the origin of the alligator that ended up as one of their straps, and not only do they not always know, but their heads are often buried in the sand [...] A skin may originate in Indonesia, for example, go to Singapore for a first tanning process, then a second tanning in Italy, before being finished in France and delivered to a LEATHERWORKER in Switzerland, making traceability an urgent priority."

I close with a pretty example of partially namelled damaskeening.






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Note added at 13 hrs (2022-02-18 23:51:51 GMT)
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Apparently it's a rare occupation, except in the Swiss watch industry, so it's hardly surprising the word 'damaskeener' is rare these days.

The collector here seems to know his stuff and is not afraid to use the word:
"Some collectors who are fans of the damaskeener's art might prefer the full-plate watches"
https://mb.nawcc.org/threads/3-4-plate-vs-full-plate.70320/