There is no known way in which nuclear radiation could have been the source of energy for the first living beings.

While a little radiation might have been beneficial at a later stage in the evolution of living beings, by increasing the frequency of random mutations, for a faster evolution, for the early living beings the only effect of radiation would have been to destroy them and prevent them for having descendants.

The only positive effect of the increased radioactivity in the early Solar System is that it is possible that not only the planets but also some smaller asteroids might have had warm interiors and volcanism.

So in none of the huge number of small bodies that exist in the Solar System there have ever been conditions for the appearance of life, but perhaps on some of the bigger asteroids there might have been such conditions, if they also had water and volatile chemical elements, besides a warm interior.

If life has ever appeared in such a place it must have used the same sources of energy that are known to have been used by the ancestors of life on Earth, i.e. gases and ionic gradients produced by chemical reactions between water and volcanic rocks.