How do 1s and 2p orbitals overlap? In the following figure  

we can see that the p-orbitals overlap 1s orbital (though relatively very little). How can an electron in p-orbital, be simultaneously in the 1s orbital at any given point in space and time?
The question is interesting, but slightly ill-defined.
If you take s and p orbitals as a solution of Schrodinger equation of hydrogen-like atom, then they are eigenfunctions of Hamiltonian, therefore are orthogonal. That means - no overlap. Do not be misled by the figure, if you do the math properly, $\langle1s|2p\rangle = 0$.

If you speak of just one electron, the solution is reasonably simple as stated in the article and the electron does not simultaneously reside in both orbitals. Even if you would think of interaction with excited states, it would not help, as single excitations do not interact with ground state. You can of course excite it, but than it is not "simultaneous".

In many-electron system, the picture gets more complicated, but this was not a question.