First let's consider a different situation. Light waves have polarization. If you imagine a light wave coming out of this screen, its electric field can be polarized vertically, horizontally, diagonally, etc., and this is also true for each individual photon.
If I pass a photon through a vertical polarizing filter, I only ever get two results: either the whole photon gets through or nothing gets through at all. So I'll only get two possible results from the measurement: vertical (gets through) or horizontal (gets blocked).
There exist materials that can rotate polarization. So you might ask, when I put a horizontally polarized photon through such a material, what is the moment when it turns from horizontal to vertical? There has to be an instantaneous jump, because it can only be horizontal or vertical, right? But that's not right at all. The polarization just smoothly rotates, through a superposition of horizontal and vertical, as we can see using diagonal polarizing filters. Just because a particular measuring device can only see two options doesn't mean only two options exist.
The same goes for your question. Now it doesn't really make sense to talk about the 'speed' of a jump because the electrons don't even have definite positions; you're just having one delocalized cloud turn into another. But the orbitals do have definite angular momentum, so you could ask how fast the angular momentum jumps. Same answer as for polarization; it just interpolates through a superposition, even though a measurement at any intermediate point will always give an integer angular momentum.
Perhaps something closer to what you want would be an electron in a double well. Starting in one well, the electron can tunnel to the other. The process is governed by the Schrodinger equation and is perfectly continuous in time. I have a feeling you're looking for a way to travel faster than light and you can in this model, but only because we're doing nonrelativistic quantum mechanics. In a relativistic theory everything would properly obey causality.