In most cases I would expect that the ascetic must deal primarily with the function of mind, so in the case of dealing with the cold, there is obviously a physical aspect of physical feeling one experiences from temperature but cold itself is just a sensory experience one can endure with patience. It is the suffering it causes, a mental factor, that the person needs to primarily deal with. Reminding themselves of the nature of samasara and the 3 marks of existence, that cold is indeed an unpleasant (also impermanent) sensation and that it is the clinging to the sensation that causes suffering, i.e the 7th and 8th dependant related links.
That being said on a more practical level. While a (traditional) ordained person is not allowed to own more than robes and a bowl, sticks, wood, tinder, kindling are not owned by anyone and enduring hardship like cold for the sake of it is an extreme, extreme asceticism. So one would expect even the sangha in the Buddhas age did in fact take sticks off the ground, rub sticks together and make fire.
Plus also keep in mind that ancient Nepal/India was nowhere near as cold as say modern day northern Europe. So if you were a European ascetic in modern times, you would need more tools and ability to make fire to literally survive, or it would be testament to suicide via negligence of oneself, again an extreme. Meaning to collect firewood and keep them dry in anticipation for the extreme weather.
Remember the original sangha retreated to caves to wade out the monsoon season. In Europe we don't have monsoons, we have extreme cold. So one would expect the same precautions of retreating to a cave (and collecting firewood) is what is needed to endure extreme cold, just like Buddha and the sangha did to endure extreme rain.
Even the dhutanghas, the ascetic practices praised by the Buddha have no mention of "enduring weather for the sake of enduring"
Then to add, in Tibet, obviously even more cold that northern Europe, specifically the vajrayana teachings, some have mastered the practice of tummo, a meditation practice gaining the ability to regulate the inner heat of the body. Where these teachings originated from I do not know, most likely post Buddhas life but it would lead one to expect such things were known and maybe practised as a direct antidote to extreme cold within the Buddhas lifetime. just of course Buddhas teachings are document only relating to the Dharma, not for yogic practises that are not directly leading to liberation.
As for resisting the urge to return to society. Well one abandons the householder lifestyle because of seeing how fruitless it is. They would have only needed to basic contemplate what life was like before going forth into the homeless life to realize that it is a folly to think reverting back to a householder life will sort the real problems out. A householder and an ascetic both get cold in the winter regardless. Plus they would still have had to re-enter society for alms round regardless.
I personally think in modern time Europe one would genuinely need support of lay people to survive asceticism like the original sangha did to endure the cold. Be it a coat, sleeping bag, windbreaking equipment, tent/coat etc or actual shelter in their home. Let alone food. One would probably get frostbite and die walking barefoot and living with only robes and a bowl like the original sangha did in modern northern Europe weather.