Is it possible for an organism to evolve to biologically generate its own nuclear power? If so, how would this evolution occur?
They already have
I'm not kidding. Candidatus Desulforudis audaxviator (alternate wiki link and PDF paper not behind a paywall) is a really curious species. It is a "single species ecosystem" which lives in deep fissures underground where D. audaxviator makes up so much of the total genetic content of the ecosystem that we have to consider that the remaining genetic load identified during analysis might be nothing but contaminants from our collection process.
D. audaxviator has metabolic pathways to process the hydrogen peroxide generated by the radioactive decay of Uranium nearby. It, in every sense of the word, derives power from nuclear sources.
There are already organisms that have evolved to use nuclear power. Radiotrophic fungus are found inside and around the Chernobyl power plant. Fundamentally, there is little difference between plants that use light radiation to generate energy than organisms that use other forms of radiation to generate energy. Any sort of energy - heat, radiation, light, or chemical could potentially be used by an organism.
It would also certainly be possible for some form of radiotrophic fungus / plant / algae to form a symbiotic relationship with other organisms like the relationship between Elysia Chlorotica and algae. Over time, this symbiotic relationship could develop to the point that the radiotropic fungi / algae become a part of the larger organism - similar to how mitochondria have been integrated as part of larger organisms here on earth.
This could possibly create an organism that seeks and eats highly radioactive waste, or basks in levels of radiation that would kill the organisms we're familiar with.
Possibly a LENR reaction would be compatible with biological processes that we are familiar with. Though widely considered impossible, enough scientists still think that LENR actually exists and is able to generate lots of energy that it would seem reasonable for a Worldbuilding scenario.
LENR reactions do not require extreme pressure or temperature and do not have the large radioactive byproducts that are so harmful for life. If LENR is real and the nanotechnology of biological processes could harvest the energy, it would seem a viable basis for a lifeform.
There is a chemical poisoning problem associate with LENR materials since palladium, nickel, etc. are used. But, perhaps a LENR reaction could be based on a suitably engineered carbon molecule. This seems like the best choice for a direct usage of nuclear power by a lifeform that I can think of.
To avoid life damaging radiation hazards you need something that either avoids the radiation in the first place - perhaps aneutronic fusion or such-like, or you must have a very powerful system of resisting and/or repairing radiation damage. Aneutronic fusion is unfortunately even more challenging than the mainstream fusion approaches. Any form of fission is loaded with ionizing radiation.
If an organism could somehow harness Pu-238 or similar isotope in the manner of a radioisotope thermoelectric generator (RTG) perhaps it use this as an energy source. Pu-238 is an alpha emitter and is quite harmful, Sr-90 is a beta emitter, but still ionizing. The RTG is a very simply nuclear design, but the radiation is still a very real problem. The simplicity of an RTG is a major plus if you could justify dealing with the radiation and the ability to renew your supply of your radioactive isotope of choice. All RTG isotopes are necessarily short half-life and not present in significant quantity in nature.
Then again, just how big can your lifeform be? If it is the size of a planet, good old uranium, thorium and potassium can generate plenty of heat and the levels are radiation are not very high -- just on that scale, the overall rate of heat loss is quite low.
Awkwardly enough, such a creature would be one that couldn't do much except channel that energy as quickly as possible. Think a creature with a drive to do something quickly, and that something would be massive, such as a zerg ultralisk from starcraft. Move as quickly as possible to get rid of heat and go accomplish a huge task, like destroy something, and then overheat and die when no objectives are left. This would be a cool weapon creature, but otherwise impossible to stop moving without overheating. Alternately, let the heat go free, like a Balrog. And it would be massively well armored to survive nuclear radiation. And... It would leave radioactive scat.
Consider the possibility of a Dyson sphere with a single organism. Since the star would be entirely enclosed within it, it's reasonable to think of the whole system as one creature. That's a single organism with a fusion plant to power it.
The single organism grew bigger and bigger until it overwhelmed (or incorporated) all other life.
For a smaller variant, consider an abandoned fission plant covered in a lichen or fungus. Perhaps over time it accidentally adds more uranium to the core. This increases the heat output. As it grows, it finds more and more uranium and puts it all in the core.
Perhaps you find these alternatives boring, but they're what I thought of given your description.
For fission, the heavy elements already exist, and are already producing heat, which doesn't quite match your description of "generate its own nuclear power". There are processes to speed up the breakdown of the elements so that they produce heat faster, but it's a bit far-fetched to imagine one of those processes taking place within a living creature. However, the overall concept of a creature eating radioactive rock and surviving off the heat they produce isn't impossible. A creature that survives on heat transfer and eats extremely heavy food would be unlike anything ever encountered on earth, so you're on your own, to invent that one.
As for fusion, the only reactions that are known to work are "hot" fusion, requiring an enormous amount of energy to start, and in turn, produce too much energy to control (but they're getting close). It is pretty unbelievable that any creature would ever be powered by hot fusion. However, there are also a lot of folks who think "cold" fusion might be possible, where just a few hydrogen atoms somehow are merged and produce a little energy, which can be controlled. How and why the atoms combine is still a matter of science fiction.... but that makes it fair game for a science fiction story like you might be writing :-)
If you write a story about strange little fish which are hot to the touch in arctic waters, eat almost nothing, have short lifespans, cause strange illnesses to anyone who eats them for too long, and produce a layer of air when kept in sealed tanks which changes your voice if you breath it, a clever sci-fi reader will realize you're describing a cold fusion powered creature.