People tend to vastly overestimate the impact of radiation and fallout. Even the by a very large margin worst nuclear disasters, Fukushima and Chernobyl, have not killed more than 100.000 people. And especially Chernobyl, with the reactor core being openly burning while melting down, generated immense amount of highly radioactive ash and dust that was deposited in the area.

Nuclear weapons, especially thermonuclear devices which have been the mainstay since the 1950s, excel at using almost the entire nuclear fuel provided to generate raw energy. There simply remains extremely little radioactive material that isn’t fissioned or fused into stable compounds in very short amounts of time during and immediately after the explosion. There’s always gonna be some grams of material with longer half lives, but as others correctly pointed out: the longer the half life, the less radiation per minute it generates. And even this remaining radiation can easily be contained when the weapon is detonated underground, which was the norm after the 1963 partial test ban treaty.