We begin this with a minor Frame Challenge
The Temperature went down 1.5 degrees
All these towns have populations numbering in the low hundreds, maybe
50-400 people at most in any given town.
This implies a much smaller nuclear war than a full fledged slugfest between a couple of superpowers. When nukes detonate near the ground, they release a LOT more carbon ash than the upper atmosphere airbursts and under water tests that mostly made up last century's nuclear arms tests. According to this National Geographic article, that amount of cooling could happen with at little as 100 15-kiloton warheads.
To see this little climate change, and that much depopulation, you need to make your victim to be nation of a very specific size. A 15kt nuke will only kill most people in an area of about 4km^2 from the blast, but it create enough fallout to fatally poison most people in a 700km^2 area downwind within the next few days to weeks.
This means you need to pick a country that is just on the edge of this threshold: enough to have a few survivable zones, but no where that most people won't die anyway (60,000-80,000 square kilometers depending on wind conditions). According to https://www.worldometers.info/world-population/population-by-country/ there are 8 countries that fit this profile: Latvia, Lithuania, Sri Lanka, Ireland, Georgia, Sierra Leone, Panaman, and the Czech Republic.
Some of these nations though have such high population densities though that it would be very hard to create such low survival rates without everyone getting deathly ill and disfigured first; so, that should probably rule out Sri Lanka, Czech Republic, and Sierra Leone.
Something I forgot to say: These people cannot go to the Southern
Hemisphere at all. All solutions must occur in the Northern
Hemisphere, as in my story these people are going to be the ones to
recolonize the continental US.
A smaller "Regional nuclear war" kinda violates these conditions, because your first world nations would be well enough intact that any survivors would just be evacuated by humanitarian efforts.
To resolve these issues in a way that works for your story, I'd suggest switching to a much deeper nuclear winter caused by a war between superpowers.
The most devastation you'll probably see in a nuclear war between two countries would probably be if the US and Russia decided to trade blows. According to official counts, the US has 405 ICBMs and Russia has 286. Each nation is also estimated to have between 300-1300 additional shorter ranged ready-to-deploy nukes. Both nations have much larger stockpiles of nukes, but most are stored in a disassembled state or designated as alternate warheads for the same missile system; so, they will mostly not be deployed in all likelihood. Modern deployable nukes average 150 kilotons, but are mostly H-bombs, not A-bombs; so, while they are more destructive, they produce way less radioactive fallout than a same yield H-bomb. The Hiroshima 15kt bombs I referenced before used 64kg or Uranium, but an H-bomb only requires ~10kg to trigger a fussions reaction; so, while the average blast of these weapons will be 12km^2 the lethal fallout zone should only be just about 110km^2. This means that if the geographically smaller United states were hit with the maximum estimated payload of Russia's ready to deploy arsonal, you'd be looking at direct hits not targeting any towns smaller than ~5-20 thousand, and fatal radiation covering about 1.7% of the nation's landmass
It is really hard to say how much cooling this would cause, because of the number of variables at play, but a 7−8°C drop seems to be the leading theory.
Initial survival rates from the radiation and blasts will leave most towns in the 5000 and smaller population range intact, but the winter would be intense enough to cause total crop failures resulting in massive depopulation of rural areas. Only people with 1+years of stored food with the ability to switch to crops meant for much colder regions would survive. 90% starvation by crop failure and fighting over remaining resources might bring your down to your target populations.
Life after Armageddon
The Nuclear Winter
Don't just think about how one will survive, but who will be left to try to survive. In a nuclear war, you make urban centers your primary targets. This means city dwellers will be eliminated from the population instantly, and suburbanites will be irradiated and burned so badly that they will pretty much all be dead shortly there after.
Those who survive will mostly be your rural populations. Most farms produce and store way more food than they consume; so, while the nuclear winter would kill off local crops, many of these communities will have massive silos with years worth of food in them. The first year will be the end of most plants and wildlife. But, farmers could adapt by planting northern crops like potatoes farther south and achieve long term sustainability.
After a few decades though, the nuclear summer will come which will cause a massive spike in global temperatures due to holes in the ozone layer. With all of our industries that actively destroy the Ozone now in ruins, maybe that part will mostly balance itself out, but if not, even more famines happen as global warming spikes up to several degrees above current levels and farmers again have to adapt new crops year over year.
Fallout and UV Radiation
Fallout will not be at fatal levels throughout most of rural america, but it will still make people sick. Farmers know more than the average person about what kinds of plants can be used to detox your body; so, rather than relying on pharmaceuticals to treat heavy metal poisoning, they may just become reliant on eating things like milk thistle seeds, dandelion root, and barley to help thier bodies purge naturally of the fallout. Thistle and dandelions are both common weeds and pretty resilient to weather change; so, many farmers will still have access to it whether it is thier intended crop or not.
IMO, UV radiation from holes in the atmosphere will be your biggest problem, but UV radiation has very little penetrating power, it can be be stopped by most standard housing materials (including window glass), and just wearing full coverage clothes blocks out most of it. If farmers adopt something similar to Chinese beach suits as the standard work attire, they will endure the UV radiation just fine.