Why electrons in low lying levels of individual atoms stay localized in their own atoms in a crystal? Doesn't this contradict Bloch's theorem?
They do not stay localized actually. It's only that their eigenstates (which are non-local) are very close in energy, and we can choose the basis of localized states and they would be close to eigenstates with high accuracy.
You can consider the tight-binding model, and take it to the limit of zero overlap of orbitals of neighboring atoms, and zero overlap integrals. That would be close to what low-level electrons in a crystal behave like.