Trying to develop a fictional scenario with a 'believable' theory as to how a Nuclear War and resulting Nuclear Winter could turn into a millennial scale ice age. By 'believable', I mean that it passes basic Science Fiction rules, but not necessarily rock solid Hard SF rules. If it's completely impossible, then is there a scenario where prolonged or periodic use of nuclear weapons over say a hundred years might provide sufficient forcing to make it happen.
There is some reason to believe that the world was heading into an ice age when the industrial revolution started. The industrial revolution of course added a great deal of carbon dioxide to the air. If you are willing to believe that this prevented the ice age, nuclear winter combined with an end to burning fossil fuels could lead to the previous ice age resuming.
To repeat, the hundred years part would be caused by ceasing to burn fossil fuels (presumably because the nuclear war killed off enough people that it was not feasible to continue the drilling, transportation, and refining tasks). It would be triggered by the nuclear war. The actual nuclear war would cause a few years of cold. But the real cold snap would hit as the world reabsorbed the excess carbon dioxide.
Skeptical Science says that human activity produces a net 29 gigatons a year of carbon dioxide. And everything else removes a net 17 gigatons (
439 - 450 + 332 - 338) from the air. So removal of human activity would switch the world from adding 12 gigatons per year to subtracting 17. So two years would take us back three in terms of carbon in the atmosphere. A hundred years after the war, the Earth might be back to where it was a hundred fifty years ago: in the Little Ice Age.
Two years of nuclear winter followed by a warmer period starting a century of cooling temperatures as the industrial age's global warming reverses followed by a millennium of ice age. If that works for you, it seems reasonable at the science fiction level, particularly as our ability to model climate is still rather weak. Since we don't know how things do work, it's hard to say that that's not how it works.
It's also possible that we could tweak this up a bit. Because nuclear winter works by throwing dust in the air, which reflects light. When that dust settles, it will sometimes cover up plants. Those plants decay underneath the ground and start the process of creating new fossil fuel. Meanwhile, new plants grow on top of them, pulling carbon dioxide out of the air and releasing oxygen.
The heat from the nuclear weapons might cause methane in the atmosphere to ignite. Since methane is a worse greenhouse gas than carbon dioxide and water vapor, this might cause some cooling.
The world might become dryer. Because the two years of nuclear winter will reduce the amount of evaporation from oceans and lakes while the colder weather will cause the ice caps and glaciers to grow. Larger ice caps and glaciers may reflect more light and while plants won't grow on top of them, they'll likely keep the plants that go under them from rotting and releasing carbon dioxide. Dryer means fewer clouds holding temperature in at night.
It's not ridiculous to think that these effects might accelerate the cooling century.
If you simply want an ice age, you may not even need the nuclear winter. Simply let the Sun enter a 1000 years long period of slightly lower intensity. It could be large enough to counteract anything humanity does to heat up the climate.
We have evidence that the Sun's output is not constant over large timescales. It is theorized that such low activity was partly responsible for earlier ice ages.
The Sun is still far from fully understood. Predictions about its solar cycles are not 100% accurate. "Your" Sun can enter an unforseen, low intensity period.
For more in depth information on solar cycles:
Hit a large gypsum, or other sulfurous compound, deposit. The main reason the that Chicxulub impact caused long term climatic chaos wasn't the initial rocky impact ejecta, they were bad but what caused hundreds of years global temperature oscillations was that the impact happened in an area rich in carbon and sulfur bearing rocks, mainly limestone and gypsum, and it pumped geologically significant quantities of both Carbon and Sulfur oxides into the atmosphere when it vapourised those rocks. If nuclear weapons were to burn off extremely large quantities of sulfur compounds and create a lasting blanket of sulfur aerosols then an ice age is a distinct possibility.