Ugh I wish I still had my copies of Of Mice And Magic and the Smithsonian Book of Newspaper Comics, those are invaluable references for this sort of question. If you are at all curious about the history of cartoons and comics, GO FIND THESE BOOKS. In physical form if at all possible for the Smithsonian one because it's full of large reproductions of lots of early strips.

I instinctively wanted to snark "as soon as people besides Winsor McCay started animating" but it turns out that some quick googling of Bray Studios characters shows a bunch of four finger types. (Bray was the first major American animation studio; they invented things like "putting characters on transparent plastic sheets on top of static backgrounds".)

Some googling at various early shorts is giving me a ballpark figure of 1920, which feels about right - people were starting to really make a business of cranking the things out, and every line you can drop in a character design is that much less time spent animating them.

Hmm, Steamboat Willie (Mickey's first appearance in 1928) has three fingers plus thumb, whereas an Oswald the Lucky Rabbit short from 1927 (Walt's previous flagship character, who he lost when he split with the company he was producing those shorts for) has four plus thumb. So tentatively I'm pegging it at 1928, though I haven't had a look at the output of other studios as I'm in bed on the ipad. I'd definitely want to see when the Fleischer studios shifted from four fingers to three. (I don't count the thumb as a finger, maybe I'm anal.)

Basically I'd say 1925-1930 for the entire industry to shift over for cartoony cartoons, and comics artists quickly following. Especially since there was a lot of crossover between animators and comic artists.

The convention was certainly in place by the 40s; all the WB and MGM shorts had three-finger hands, often in little white gloves. Which probably became a convention as a way to make a B&W character's hand "read" (animator jargon for "be seen") when in front of their body.

Also I would like to link to the insane animation of Zig-Zag the Grand Vizer in Richard Williams' "The Thief and the Cobbler", who has SIX fingers and a ton of rings, but I can't find it offhand and want to go to sleep.

Not a historian, but I'm an ex-animator who obsessively pored over the history of animation while she was growing up. Pretty much every animator I've ever met is a bit of an amateur animation historian!

(And of course it's also a convention of AMERICAN animation, it's never taken hold in Japan due to association with gangsters chopping off fingers when they make major mistakes.)