Interesting stuff. I think there may be some aspect of cross purposes/misunderstanding here? It seems that going by these newspaper snippets a fashion arose for men in the US to start wearing fob watches in "straps" designed to hold them. Enough of a fashion for it to be reported anyway. As the article notes the British military types had started to do this(reports and photos from the Boer war show this) and the American lads took it up. So it's entirely possible, if not a given that many of these watches carried in those wrist conversion straps were of local US manufacture.
However, while they were watches carried on the wrist*, they weren't wristwatches. They weren't designed, made or sold as wristwatches by the companies involved on either side of the Atlantic at this stage. They were end user conversions. It would be akin to me buying a brand new Panerai or whatever, cutting off the strap, attaching a chain to it and calling it a pocketwatch. It might function as one, but it wouldn't be one.
For the first real Man's wristwatch the definition would surely be one designed and made from the get go for the purpose. In this case the Omega's Horologist007 referenced would be such a beast.
Now I have read in a few places that Girrard Perregaux had built a batch of wristwatches for the German navy before this time. The 1880's I think? That's the claim, but I've never seen any documentation, never mind an actual example of one to prove it. On their own website a few years back the example they had was clearly from circa 1916, not the 1880's. For pre Omega examples I'd reckon a long perusal of photos from the Boer war might turn up an actual wristlet, though making out who made it would be next to impossible. I have seen the fobwatch bracelet in pics from that time alright. I recall reading of a letter from a guy in that conflict who wrote about an engagement where his wristwatch was specifically mentioned.
From this page
Anglo Boer war musuem
(first pic on the page has a Canadian bloke sporting one of these fobholderwatches in 1899)
Halfway down the page we have a letter from a Canadian chap by the name of Otto Moody in 1902(though possibly earlier as he signed up two years after the previous soldier on the page and that guy was seeing action in 1900). In it he says "I wished you could manage to get me one of those small watches with a watch belt to go around your wrist like this(he draws a picture of it). Most every fellow here has one, they are very handy". The tiny doodle he sketches such as it is does not resemble a fob watch in a holder, but an actual wristwatch as we might recognise it. Given that "most every fellow" had one it seems that by 1902 anyway they were not a rare item in such a setting, they could be bought in Canada as he asks his mum go source one and that they were likely being manufactured by
someone. The numbers made, by actual manufacturers, or privately made small runs by jewelers would be questions though.
Personally speaking and it's just musing on my part, but I would bet the farm that there were small runs of these wristlets/wristwatches in the 1890's and sooner or later one or more will show up.
*as women were already doing. Consider the article written about the guys in Cuba sending home for "leather bracelet cases". Clearly these already existed and most likely they were for women.