This one is complex. There was a piece called March of the Tin/Wooden Soldirers from 1897, but the thing people associate with "March of the Wooden Soldiers is actually the 1903 Victor Herbert operetta Babes in Toyland, which was a Christmas-themed extravaganza that featured a tune “March of the Toys” and also wooden soldiers.
In 1934 Hal Roach produced a Laurel and Hardy vehicle very loosely based on Babes in Toyland, and called by that name, although it became better known as March of the Wooden Soldiers. as far as I know, it isn’;t related to the Leon Jessel music, but I could be wrong. The Wikipedia page doesn’t mention that it was used in the movie. The film featured a rescue, at the end, of Toyland from an assault by the Boogeymen when Laurel and Hardy’s toy soldiers (instead of producing 600 1-foot-tall soldiers, they made 100 6-foot-tall-soldiers) are activated and rout the invaders. There’s a great stop-motion scene of the soldiers marching out, although for the rest of the movie they’re played by people in soldier costume with half-circles attached to their feet.
(Disney was a fried of Roach’s, and let him use the character of Mickey Mouse, played by a monkey in costume)
Since the film, like the operetta, was Christmas-themed, it always got shown around Christmas time, especially many years later on television when it was 25 or more years older. It got a new lease on life when it was computer colorized in the 1990s.
To tell the truth, though, there’s no explicit reference to Christmas in it that I can recall, or that the synposes online say anything about.
Disney made his own version of the story in 1961, starring Annette Funicello, Tommy Sands, Ray Bolger (!! The Scarecrow of Oz himself!) as Barnaby, the Bad Guy, and Ed Wynn as The Toymaker (three years before his role in Mary Poppins). He removed just about all the references to Christmas, except for the Toymaker being pressed to finish his toys by Christmas. Nevertheless, the fil m was released at Christmas, and Disney uses Toy Soldiers from the movie in Christmas parades.