To start, you need to select one of the "Winter" maps. They are Snowy Coast, Icy Islands, and Frosty River.
When it actually starts to snow might vary a bit depending on how fast you play and some other unknown variables. The first snowfall I experienced was while playing on the Snowy Coast map. I was playing at 3X speed, and it started snowing immediately as I hit "Boom Town" level, which is a population of 2400 (I don't think that was a coincidence). In real time, I was playing about an hour, but again that will change depending upon how fast you build up your town.
Some experimentation might be needed, but I'm fairly certain that the appearance of snow is related to your population level, since this is when heating buildings become available.
Edit: So I started up a couple of cities and didn't touch anything, putting the speed at 3X. The time it snowed was random but averaged about 15 minutes. Again, I don't know if there are some variables that affect this, but be sure that it will snow if you select a winter theme. Just be patient. This is confirmed in comments by JumboWario. Thank You.Please provide feedback if you've seen differently.