While a Nest Cam (battery) will continue to be able to run off of a wired connection in the cold, Nest Doorbell (battery) cannot charge its battery in the cold and once that battery dies, it will stop working entirely until you warm it back up.
The new Nest Doorbell (wired) has all of the same features as the new battery model and for the same price but works just fine in cold temperatures with a minimum operating temperature of -4°F (-20°C) and no charging to worry about.A traditional doorbell combined with a Nest Cam (battery) would be a good solution if you don’t have the choice of using a fully wired doorbell but can have a camera plugged in nearby.
Related: Nest Doorbell (wired) review: A solid upgrade waiting on a better Google Home app