Guessing massive numbers like the number of flakes in a snowstorm, the number of grains of sand in the world, the number of blades of grass and the like is all about approximation. So I'm gonna dive right in here and make loads of wild guesses in the hope of getting an answer that nobody would dare call incorrect. Sound okay?
Okay, let's assume for the sake of argument that we're dealing with a storm lasting about five hours, and that it's about the size a largish city. I'm gonna go with my home town of Nottingham, which my atlas tells me is very approximately a 6.4 mile square (it's about the same size as one of the big squares). I like metric measurements, so that's (thanks Cyt) a 10.3km square so call it 100 square kilometres. We'll also assume the storm is reasonably light, so about ten snowflakes landing per second per square metre of ground. Dunno how much snow you get in your many and various countries.
Five hours is 5 x 60 x 60 = 18,000 seconds. I'm feeling generous, make that 20,000 seconds per storm.
100 square kilometres is 100 x 1000 x 1000 = 100,000,000 square metres.
10 snowflakes per second per square metre multiplied by
100,000,000 square metres multiplied by
20,000 seconds per snowstorm
equals
20,000,000,000,000 snowflakes per snowstorm.
Twenty trillion. I dunno about you, but that sounds kinda low to me, but the answer is of course give or take three orders of magnitude, or maybe more. Feel free to correct me if you have the faintest clue what you're doing.