The effect of nuclear radiation is not fully understood in real life, so it would be hard to extrapolate to your rather interesting world. The current theory is that chronic radiation exposure is cumulative, accumulating through your entire life. There is also a concept of "acute radiation syndrome," which is what happens when your body receives a sudden influx. They start roughly around 100 rad and progress in lethality up to around 1000 rad (a few conversions suggest that 100 rad is roughly 20 years worth of permitted radiation for those who work with radioactive materials). Above that point, people just die. Nuclear weapons can release a large amount of radiation up front, so this may cause acute radiation syndrome no matter how much your humans were used to radiation.
Now the real question is how your humans protected themselves against radiation. A few thousand years, evolutionally, is a blink in time. It's virtually nothing at all. It is highly unlikely that we would evolve meaningful anti-radiation protections over a few thousand years, even in such a hostile world. We would more than likely just die. If we did develop a protection, it would be impossible to answer your question because how we end up protecting our genome will matter greatly. Without you specifying exactly what chemical mechanism is being used, we can't really help there.
There are some interesting edges to explore here. One key unknown in radiation studies is the effect of small doses on the body. We know that the amount of damage done by medium to large amounts of radiation is directly proportional to the amount of radiation received. In fact, this is so linear that we have units like rem or seiverts which are designed to track cumulative radiation effects on the body. However, it is not yet known whether small amounts of radiation still have this linear effect, or if they are actually easier to repair. We just don't have the data. If they are indeed easier to repair, there may be a mechanism you can leverage here to repair larger errors. It's much easier to re-balance an existing system than to evolve a new one.
One solution we could look at is genetic manipulation. Perhaps humans decided to change themselves to be more radiation tolerant. We might put checksums into our DNA to help us correct for errors. Such checksum like systems exist in our body today, but they're organic constructs that are designed to do "just enough" to keep us growing and evolving. A synthetic checksum could lock down our genome far greater. However, this would also create its own issues worthy of a novel.
The other solution would be to simply hide in a hole. The easiest way to survive radiation for a thousand years is to simply go somewhere that the radiation isn't found, and stay there. In such a case, obviously a nuclear bomb would have its full devastating effect.