(So, I am going to assume you mean atomic bombs, not hydrogen bombs. H-bombs — which used nuclear fusion — were invented after Hiroshima. Atomic bombs — which use only nuclear fission — were what were used on Japan.)

So I have not read this particular book, but one could imagine two kinds of claims discussed in the 1940s. One is whether there were significant radiation casualties at Hiroshima and Nagasaki. This was controversial in 1945 because the Japanese claimed there were, and the Americans initially claimed there weren't, on the basis that anyone close enough to suffer significant amounts of radiation (enough to induce acute radiation syndrome) would have been so close that they would have been killed by the other effects (blast and fire). It was discovered within months of the end of the war, after American scientists and physicians went to Japan, that the Japanese were correct: there were a significant number of people, maybe 15-20% of victims, who suffered from primarily radiation-induced injuries. See more discussion of this here. Any author who denied this in 1949 would have been flying in the face of much published research by the US and Japanese governments.

The other position is about radioactive contamination. This is a long-term exposure to chronic radioactivity, what is called "residual radioactivity" as opposed to "acute" exposure. It is the difference between saying, "the Japanese people were exposed to a lot of radioactivity all at once" and saying "the cities were made radioactive."

In the immediate aftermath of Hiroshima there was at least one major case of a scientist claiming that the use of the atomic bomb would render the city uninhabitable because of the residual radioactivity. This was based on the assumption that significant fission products (the split "halves" of the atoms used in the nuclear reaction) would collect on the ground near the explosion. This can indeed happen (though the effects are not quite what this particular scientist thought), but did not at Hiroshima and Nagasaki: the bombs were detonated at a very high altitude, so their fission products never mixed with dirt and debris and was carried away from the cities for some time. This allowed the worst of the by-products to decay away, and when the by-products did "fall out" of the cloud, they were also considerably diffused, and so no individual area of land got very much residual radioactivity. Later studies have confirmed that pretty much all of the exposures that had any health effects were fairly immediate, and not residual, except maybe people entering the city within a day or so of the blast.

There are occasionally people who think that Hiroshima and Nagasaki were contaminated, or that there was significant fallout as results of the bombings. This is not the case. If the author is speaking to this point, then he is not totally wrong.

If the author instead is saying that the bombs do not have radioactive effects at all, this is just false and would have been seen as false by anyone who read the literature on the subject at the time.

Again, I do not really know the authors' claims. Googling the text makes it sound like a very idiosyncratic travelogue.