How do you deal with flats in the winter?
Prevention (good tyres, good inflation, regular checks). 

For most rides, walking isn't an option, while I do sometimes run my commute, I use the bike because generally I want to get home. So a spare tube and CO2 is obligatory - with practice and a decent place to work (which can't always be guaranteed at the side of the road) I can change a tyre in less than five minutes. 

In colder weather (at or around freezing) I ride with thin liner gloves inside cold weather gloves and I find I can change a tyre with them still on; this both keeps the fingers slightly less cold, but also a little cleaner. 

If you're operating in a climate that regularly sees weather that's cold enough that five minutes without gloves is not an option, chapeau. But I think I would be looking for run-flat options (e.g. goo) and some way to get me home quicker than on foot!