We're all giving you basically the same answer, but to be more specific...
Use L. Ron Hubbard's Solution
In Battlefield Earth in order to create a crisis for the story, Hubbard placed the village from which the protagonist hails near a nuclear mine field.
You see, radiation isn't an illness and it's not contagious. You're in danger when exposed to a source. You're not when you are not exposed. Thank goodness for the Van Allen Radiation Belts!
Anyway, Johnny Goodboy Tyler's Psychlo-driven education led him to realize what the problem could be and a map he found at a defense base showing the location of the mine field, then he used Psychlo breathe-gas (which is violently reactive to radiation) and his horse to run back-and-forth and identify the exact locations of the mines. After marking them, he proceeded (with all due narrative nonsense) to relocate his kith and kin someplace safe.
Thus, your one and only option is to use something radioactive — like radioactive land mines — to rationalize the presence of radiation which affects the geographically-unfortunate population.
Why does everyone want to do this 500-1000 years after the fact?
The more you rely on physics, the more this isn't going to work. Radioactive substances have a half-life, meaning that each time the time period expires, half the strength of the radiation remains. If the half-life is 100 years then 100 years after the apocalypse the radiation is at 50% of strength. Another hundred years and it's at 25%, then 12.5%, then 5.75%... After 1,000 years there's so little radiation that there won't be any radiation poisoning. But this is looking at an enormously worst-case solution that doesn't explain why Hiroshima and Nagasaki are inhabitable today. So why are they? I found a good explanation on Quora:
First, the bomb core was about 20% U-238 which is the same stuff as in the granite countertops in my kitchen. Radiologically speaking, such materials are nearly harmless, as the vast energy stored in their nuclei is being released over an equally vast period of time, and in the form mostly of alpha particles, which are utterly harmless unless inhaled or swallowed.
In fact, the potassium-40 in your bones is much more dangerous—radiologically—than natural uranium, and about as dangerous as the enriched U-235 used to make the bomb. Worse yet, it’s inside you (and inside every living thing on Earth). Which should tell you something.
Every minute of your life, potassium-40 inside your body undergoes a few hundred thousand atomic disintegrates. All of these emit radiation which can kill or damage nearby cells. A few of these even emit photons of gamma energy, which are even worse. Yet, we’ve gotten this far.
How? Because we are evolved to handle it. Our bodies routinely repair such tiny insults from radiation, and from field studies we know that living things—including people—can safely accommodate much, much higher exposures that that. We set exposure limits in industry low because someone is ultimately responsible for it, but nature has set its limits much higher—because it has to.
The main danger from the uranium in the bomb was that it is a chemically toxic metal, but blasted to hell and spread to the winds, it quickly reacted with other chemicals in the environment or joined the millions of tons of natural uranium already dissolved in the Earth oceans. It’s harmless. Forget about it.
No, the danger from a fission bomb comes from the flash, the blast, the instantaneous burst of radiation at the moment of fission. At Hiroshima, that burst of radiation was in the form of x-rays and neutrons. The x-rays were all absorbed by the air to create the fireball, as the weapon exploded nineteen hundred feet above the ground. The neutrons, however, could reach the ground and there crack atomic nuclei, but mostly they had that effect on the bomb components, rendering some small amount of mass intensely radioactive.
This material, in the form of extremely short-lived isotopes, then went on breaking down and releasing large amounts of dangerous radiation until, within a week it was essentially all gone. After that, there were a few isotopes remaining with intermediate half-lives, some of which were dangerous to humans. In particular, radioactive iodine was a threat mostly because it can be concentrated in the thyroid and cause cancer, though this won’t happen to people who already get plenty of iodine in their diet (and most Japanese do). Next was strontium-90 which can be taken up in the bones and cause leukemia, as it is a chemical analog to calcium.
But the Hiroshima bomb barely worked. It fissioned less than 2% of its bomb core, producing relatively little radiation. All the hot stuff was gone within days. The iodine was effectively gone within months. The strontium was effectively gone in a few years. And today, there’s functionally nothing left.
Hiroshima is still radioactive, but so is New York. And so are you. Essentially all of the Earth is radioactive if you’re willing to set the bar low enough, but we can take it. We evolved here, and radiation is not magic death cooties. Meanwhile, most human exposure to unnatural radiation is from radon released by coal fired power plants, and pollution from such plants kills tens of thousands of people per year. (Source)
And that's why I say use the L. Ron Hubbard solution
Forget reality. Reality is boring. In a real nuclear apocalypse the vast majority of the Earth will be completely inhabitable again in 7-12 years and we need just enough people to survive those 7-12 years and we're in good shape... for having been stupid enough to elect governing officials who are, themselves, stupid enough to launch nuclear weapons. But I digress....
Come up with something clever, like nuclear land mines... something that supports your story and helps you with your narrative, and use that window dressing to rationalize the rule for your world that you already have: radiation is still harming people based on geographic location after 500-1,000 years.
