So just how do you get caulking to work outdoors when you have put the job off so long that the cold weather has caught up with you?
First, read the caulking tubes to see under what conditions any particular caulking can be used. Polyurethane and most Thermoplastic caulking should be used at temperatures above 10 deg C (50 deg F). Silicones and some Thermoplastic can be used all the way down to -30 deg C or even -37 deg C (-35 deg F) -- that is if your hands are still functioning.
Whether you are trying to cheat the temperature or it is just cold out -- here are two tips to help make everything work better.
Keep your caulking inside the house at least 24 hours before you use it, and bring it out one tube at a time. That way at least the caulking itself will be warm and workable.
If the surface is cold, use a hair dryer or hot air gun and run it just ahead of the caulking tube to warm the surface where the caulking will be placed. This will not warm the wall up deep down, but will allow for a better surface grab before it gets cold again.
By the way, caulking on the outside of the house has only one purpose -- to keep out water. This does not stop the wind and does not stop cold air drafts into the house because almost all siding is ventilated specifically so that the wind can get behind the siding to keep everything dry. Stopping the wind, or air sealing, is either done with a sealed house wrap under the siding during new construction -- or done by weather-stripping and indoor caulking in an existing house.