What happens if someone would make a wire coil around nuclear reactor core? What are the possibility to capture radioactivity directly?
An electrical potential is generated in a wire coil if there is a changing magnetic field through the coil. However a nuclear core doesn't generate magnetic fields at all, let along of the type required to generate electricity. So all the wire would do is get hot, and eventually become radioactive.
You might be interested in my answer to Can we make usable energy from subnuclear particles? as it is somewhat related. Basically the only useful energy we get out of a nuclear reactor, or any radioactive source, is heat.
It seems like there is confusion regarding radioactivity and radio waves. This is a coincidence in naming, not a physical connection. My brief searching shows that the radio in both cases is from the same Latin that gets us "radius," and it comes about because in both cases stuff is emanating away from the source (radially in fact).
Radioactivity amounts to either particles (alpha, beta, or neutrons) or high-energy photons (gamma rays). Gamma rays comprise the highest energy classification of photons. In general, high-energy anything, whether particle or light, only "sees" small structure. Whether an incident gamma ray or alpha particle is absorbed by a piece of material has nothing to do with how that piece of material fits into some larger structure (e.g. a small part of a coil or wire doesn't care that it's part of said coil when it comes to interacting with these things).
Radio waves, while also photons, are at the complete opposite end of the electromagnetic spectrum. They are extremely low energy. Large conductors (antennas) are good at picking up radio waves because they are approximately the same size and thus resonate at radio frequencies. Actually, the coiled wire found in primitive radios is used for tuning to particular frequencies, not generally for picking up the signals to begin with.