(Slightly) Weird Science
These stories were written in the 1930s, a period when the nebular hypothesis, the idea that the planets formed by gradual accretion of gas and dust particles, had fallen out of favour. Solar systems were considered to be unlikely to form without outside influences. The theory then in vogue and used by many SF authors assumed a more violent origin for planets, in which another star passed close to the Sun and gravity pulled streams of gas from both of them to form the planets. The consequences of this theory included rapid planet formation and a more even distribution of elements through the solar system. Gas giants would still radiate internal heat from their formation, warming their moons sufficiently for them to be more or less habitable, and there would be radioactive materials on their moons.
The result is a younger solar system in which there is life on every world, and in which humans can survive with minimal protection on Venus, Mars, and the moons of some gas giants.
Nuclear processes work differently in this setting. Radioactive rays cause the atomic nucleus to break down, releasing huge amounts of energy. The more powerful the radioactive source, the heavier the elements it can break down1. This is slow and controllable process, making atomic rocket engines possible. Their exhaust is hot and immensely destructive, but doesn’t appear to be dangerously radioactive.
Nuclear materials are surprisingly stable; for example, commercially viable deposits of protactinium exist naturally on Europa, in our universe most of its isotopes are extremely unstable and it is only found in traces as a decay product of uranium.
Atomic bombs exist, but they presumably involve a more destructive reaction which breaks down the radioactive material itself.
1 Redemption Cairn