Much has been claimed in regard to who 'invented' or popularized wristwatches. Notably, Breguet built one in 1810, but I have seen bracelets with small dangling watches that go much farther back, the 16th century. These were designed for and worn by women.
There is an often repeated claim that the soldiers of WWI were the first to use and popularize the wristwatch, but this is incorrect. Girard Perregaux marketed military wristwatches for the German Military in the 1880's, and wristwatches were popular among Soldiers of the Anglo Boer war.
Spanish Soldiers likewise used wristwatches in the 1880's.
As to whether this 'caught on' when the men came home is up for debate. Wristlet watches or bracelet watches were seen as feminine well into the 20th century, especially in North America and the UK.