how long does nuclear radiation stay in the air

It's not really like smoke that just hangs around and eventually clears out on its own. Depending on what kind of material we're talking about, the radioactive particles can stay suspended in the atmosphere for a few days if they're light, but they usually settle onto the ground or get washed away by rain pretty quickly. The real issue isn't really the air itself, but where those particles end up landing.

Once those particles settle, they stick to everything like dust. That's why you hear about people having to wash down roofs or scrape off topsoil after an accident. It stays in the environment for years, even decades, depending on the half-life of the specific isotopes involved. It's definitely not something that just vanishes into thin air after a breeze blows through, which is the scary part if you think about it too much.