A 390-year-old bonsai tree survived an atomic bomb (washingtonpost.com)

There were many acute radiation poisoning deaths (though there were far more deaths from the fire and the blast), and about a third of the survivors suffered radiation injuries that increased their risk of cancer, but there was very little to no long-term contamination of the area.

Ground blasts are worse, but it's the ground itself that becomes radioactive, not the things that are far away and are affected by the blast.

And no one purposely does ground bursts, because they don't do a good job of damaging things with a blast - most of the energy ends up reflected into space, or absorbed into the ground. (I guess a terrorist might want to, since they don't have military strategic goals.)

The material of the bomb itself is radioactive of course, but there isn't usually all that much of it, and it's dispersed quite widely.

Her journals did not become radioactive because they were neutron-activated, but because they were contaminated by radioactive material. But there's a limited amount of that, so only a small amount of material becomes radioactive that way.

(Note: I'm speaking relatively here, if you have a sensitive enough detector you'll find stuff of course. That's one of the things that makes people so scared of radiation - it's so easy to detect at even minuscule levels, while far more dangerous things are harder to detect so no one reports them.)

In terms of the nuclear blast and radiation itself, it's true that probably only a half a billion people or so would die. But you'd be essentially ripping out the entire economic and political nervous system of the world, resulting in massive economic dislocation and political chaos. I find it plenty plausible that that by itself could easily get the body count up to the billions.

There are also possible environmental effects: the dreaded nuclear winter. It's not clear to me if the debate was resolved one way or another about how bad it'd actually be, but it seems like a real possibility that it'd hurt the climate much worse than even the most dire warnings of global warming. And it'd happen more or less instantly, as black soot would immediately be sent up to the stratosphere.

An article from 2007 in the Journal of Geophysical Research, courtesy of Wikipedia, states:

    A global average surface cooling of –7 °C to –8 °C persists for years, and after a decade the cooling is still –4 °C (Fig. 2). Considering that the global average cooling at the depth of the last ice age 18,000 yr ago was about –5 °C, this would be a climate change unprecedented in speed and amplitude in the history of the human race. The temperature changes are largest over land ... Cooling of more than –20 °C occurs over large areas of North America and of more than –30 °C over much of Eurasia, including all agricultural regions.

The math started with WWII Germany. They kept reasonably reliable records of where and when people died during allied air raids. So someone drew up a formula linking the tonnage of bombs dropped with the number of dead on the ground. This was then extended to nukes and their power measured in thousands (kilo) and millions (mega) of tons of TNT. So bombs with explosive power greater than all of WWII (a common description) could "kill" as many people as an entire war. Kissinger was correct in saying that all the bombs in the world would not, could not, destroy everyone ten or twenty times over.

This false analogy was also part of debates during Vietnam. Many, on both sided of every argument, used the math to estimate the number killed by US bombing raids, ignoring the fact that German cities and Vietnam jungle are in no way analogous.