Is the nucleus of an atom neutral, positively or negatively charged?
The nucleus is formally positively charged (all nuclei are!).
Protons, massive, positively charged nuclear particles, have an opposite electronic charge to electrons, extra-nuclear particles of negligible mass.
The positive/negative charge convention is entirely arbitrary. The point is that protons and electrons simply have opposite electronic charge. I have written before here that it would be easier for chemists had the electron been designated a positively charged particle, and the proton a negatively charged particle. It would have saved generations of quantum chemists who got the sign on their answer wrong, just because they counted even and not odd, or vice versa.
In the nucleus, while of course there is electrostatic repulsion between like charges, there is a short range strong nuclear force that binds protons and neutrons together.