There certainly is a statistical correlation between the position variables of the nucleus and the electrons - if the electron happens to be somewhere on the Moon, it would be unlikely to find the nucleus here on the Earth. So there is an entanglement. I'm not that sure whether the spins of the nucleus and electrons are entangled.
Many electrons in an atom are entangled with each other. I don't think it's possible to specify every combination that could occur. Possible spin entanglement with the nucleus too.
In a pair in a specific orbital, where one has spin up and the other is spin down - and indistinguishable, I would expect entanglement. However, I am not sure the entanglement would be monogamous if other orbitals have indistinguishable electrons.