Did you know that temperatures dipped in Medieval times in something close to Nuclear Winter? It was called the Little Ice Age. NW takes temps down by 1.8 degrees F. The LIA took it down by 1.5 degrees Fahrenheit. Unless you are looking to add NW ON TOP of that--gotta say, historically, you ALREADY HAVE the effects, or at least pretty close.
So Famine. And yet the population still grew as people adjusted. There's a .3 difference. That's a lot.
NW, or at least the 1.8 takes 100 Hiroshimas. That's just...a lot. And difficult to achieve for a population so low. It means that there have to be whole areas that are just burned to ash.
Considering the population of Medieval times, this means a LOT of wizards. In fact, to achieve 100 Hiroshimas, I wonder that there would be all that many people to oppress considering all the wizards needed to do this. Further, it would NOT happen all at once--Hiroshima was a one time event, as are most nuke drops. What you're talking about (massive fires and ash in the atmosphere) will likely NOT happen all at once. Medieval times were fractured as far as communication and travel is concerned--so it's not like it is now. Concerns are local. The wizards would be warring over time, not all at once. Which means that the drop is going to be more gradual than it would be if you did 100 Hiroshimas.
The amount needed to be burned is on a scale that is far over what Medieval times could achieve. Total destruction of a city is definitely not the way war was waged most of the time. Because you wanted to take territory and resources. War waged as it has been in modern times is very, very different.
The destruction of Hiroshima was about 5 square miles. With a lot of material to burn. The largest city in England, London, was maybe 1.12 square miles. And this was a mega city of the time. That's the main problem with the premise
The answer to how they can weather it--hie thee to yon google and ask about the effects of the Little Ice Age. Folks died, famine happened and so on. That can give you a detailed account.
I will add to this when I have time regarding the Great Famine and effects.
Thing one that happened was a shift in weather patterns--specifically far more rain than was good for crops. Lots of things died.
If the temperature continued to drop, this would worsen (less rain eventually, some more extreme weather).
People would die, but people would also adjust. No matter how much the technology lacked. I posit that wizards might not be blamed for this directly (cause and effect wasn't a strong suit of the Medieval mind) and that magic would be used like technology is to help US survive.
See, even if 100 Hiroshimas happened, though there would be groups that "blamed science" most humans would grasp at ANY advantage to help them survive, even the very science and tech that destroyed them. That's how they survive. A few folks with hedge magic helping their town survive and making things grow...