The Earth was made radioactive due to a plot by the Spacers. On the other hand, the Foundation - the civilization using miniature nuclear reactors - is happeni

The Earth was made radioactive due to a plot by the Spacers. On the other hand, the Foundation - the civilization using miniature nuclear reactors - is happening many millennial later and far away from Earth. So the "miniature nuclear tech" and "radioactive Earth" aren't related in the plot.