Not true.

What you describe is a perpetuum mobile.

No kind of life can exist without a continuous flux of energy. The formation of any organic polymers requires energy, it cannot happen spontaneously.

Nucleotides that collide do not form polymers in water. The polymerization of nucleotides and of amino-acids and of most other organic macromolecules is done by water extraction (a.k.a. polycondensation).

Water extraction inside water requires considerable energy (i.e. you must dry some thing while it is still submerged in water). This is provided by certain dehydrated molecules, like ATP, from which water must have been extracted as a result of capturing some energy from the environment.

The common ancestor of all living beings known on Earth obtained energy by the reaction between free hydrogen (dihydrogen) and either carbon monoxide or carbon dioxide. It is likely that the capability to use carbon dioxide has appeared later and the very first living beings were powered by the conversion of dihydrogen and carbon monoxide into acetic acid. The energy from this reaction was captured because the first result of the reaction was not free acetic acid, but acetic acid condensed with another molecule, like in the acetyl-CoA that is used today by all living beings. Than that condensed molecule could be used to extract water from other molecules, producing dehydrated molecules like ATP, which can be used for the polymerization of nucleic acids.

There is no way to circumvent the need for a continuous source of energy for the appearance of life. Besides the H2 and CO gases, there was an alternative source of energy that was used very early, but it is not known if it was already needed for the initial appearance of life or it began to be harvested at a later stage. This second source of energy is the difference in ionic concentration between the acidic water of the primitive ocean and the alkaline water that is produced by dissolving volcanic rocks in places like hydrothermal vents, where H2 and other gases are also produced. The ionic concentration gradients produce ionic currents, which can power certain chemical reactions, like they also do in all present living beings.

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.

There not only there was no known source of energy for the synthesis of organic polymers, but also everything, with the exception of interstellar dust particles, was in a gaseous state, not suitable for forming the structure of a living being.