In my city, drivers work long shifts. Usually, when Environment Canada issues a warning, everybody gets called in the office before the storm hits. Some people at the city are actually paid to watch over the weather
Some plows start by clearing the main arteries as soon as it starts. At midnight, the rest of the team is sent out to clear everything else at least once. Some drivers work for more than 12 hours straight.
The city streets get a priority number. Priority 1 is main roads and access to schools and the hospital. My street is rated at 4 (lowest in the line).
Once the snow is pushed to the side of the roads, it can be quite some time before it is actually picked up. When the streets get dangerously narrow, they send out the removal team. They start by scraping everything to the center of the street leaving enough space on one side for large 12 wheel dump trucks. Then, the big snowblower comes by and shoots everything into trucks. Everything gets carried off to special snow disposal sites where it is packed and left to melt in the spring (there's usually some snow left in June).
So, there you have it, this is how we deal with it. I must say we're all used to driving in (and swearing at) the white stuff.
When we get some more, I'll try and shoot a few pictures on the How it works... massive equipment is always involved.
As for your question about how the drivers get to work in the morning, they simply don't