It is unfortunate, but many first introductions to electromagnetism place an emphasis on what the charged particles are going and largely give the electric and magnetic fields themselves a backseat role. This is precisely the wrong way to think about electromagnetism: the fields themselves are of paramount importance. Considerations about what electrons are doing come second (and as a consequence of what the fields are doing).
I should also point out, before carrying on to explain what I mean, that energy does not "move" as it is not a quantity defined to exist at any given point in space. A better concept to use would be energy density, which is defined point-by-point throughout space. This concept is, however, more complicated than total energy because it ties closely with other concepts like momentum density and requires calculus to discuss. It also causes some confusion because the electric and magnetic fields themselves carry both energy and momentum, a point which is often not made to students till graduate coursework.
Now, in an electrical circuit, the thing that propagates at the speed of light is the electric and magnetic fields, certainly not the electric charges (though in materials light moves somewhat slower, it is for essentially all applications equal to the speed of light). The reason the current appears to "turn on" (meaning start to flow) at the speed of light is really because the fields have done so.
Consider the electric field for the moment and just to demonstrate the point, suppose we are looking at the very long wire and that the electric field is initially zero everywhere. Then suppose we do something to cause an electric field to appear at one end of the wire and begin propagating to be whatever non-zero value we set it to all down the wire. Since the electric field propagates at the speed of light, it will take some non-zero time for the other end of the wire to have a non-zero electric field.
Now let's think about the electrons, say, in the wire. Initially, the electric field is off, so the electrons don't experience any Lorentz force (other than the electric fields due to other electrons and atoms, but we will ignore these for sake of simplicity), and hence won't really go anywhere. But as soon as the electric field at a given electron is non-zero, that electron will experience a Lorentz force and begin to move accordingly. Assuming the electric field is oriented to point down the length of the wire, the electrons will begin moving down the wire producing a current by their collective motion.
So at no point does any electron move at the speed of light, or anywhere close to it. What does move at the speed of light is the "signal" (the electric field) that tells each electron to start moving.
This is also how AC current works. The electric field is changing in time, switching directions up and down the wire. So, the electrons get signals to start moving in one direction or the other accordingly. That signal (again, the electric/magnetic fields) can move at the speed of light without the electrons themselves ever needing to do so.
This also explains why, for example, a telephone (thinking land-lines so we aren't dealing with electromagnetic waves propagating through the air) can transmit the audio signal at the speed of light (for most applications on Earth, the distances are small enough that the delay due to light speed isn't very noticeable). We don't need an electron to travel from one phone to the other, what we need is to send a signal which tells the electrons on the other end how to move, and then just "read/measure" what the electrons on the receiving end are doing (then reconstruct the audio from that signal, but that's a separate problem). In effect, only the electric/magnetic fields have done any traveling. The electrons themselves are, for many intents and purposes, staying in more or less the same spot just moving back and forth according to the fields.
Edit: To clarify a few things based on the comments. The description above is largely meant just to convey the idea that the right object to think about is the fields rather than the electrons. If we are being more accurate, a wire should be modeled as a waveguide, so the relevant fields are really propagating in the dielectric encasing the wire. Within a conductor itself, electromagnetic fields experience an exponential attenuation and hence do not behave quite in the way we think of when we speak of "waves." Specifically, the frequency of oscillation becomes complex-valued.
Furthermore, let me also note that the description given above is completely independent of how the electromagnetic fields are generated. They could be the result of an actual charge difference between the ends of the wire, by the motion of some charges at one end which would cause a propagating wave, or some combination of factors.
In particular, since the OP has asked about charge differences specifically, let me note that the exact details will depend on how the charge difference comes into being. If we were to consider the simplest example in which we were to snap our fingers and bring a charge difference into being (take the charge difference as our initial condition), then we would see a non-zero electric field propagating from either side to meet in the middle. This would cause the electrons on either side to start moving before the ones in the middle.
I suppose it might be interesting to work out the exact details of these dynamics, but let me describe what I expect to occur. The non-uniform motion of the charges in the wire due to the initial conditions will cause "lumps" of non-zero charge to appear along the wire, and in particular, once the charge difference between the ends has been equalized, there will remain non-zero charges along the wire moving around.
How these lumps along the wire will move will depend, I suspect, on the self-interaction of the lumps with the electromagnetic fields they produce by their own existence, which makes the problem rather more complicated and possibly even demanding renormalization techniques (see, for example, the self-interaction calculation for a point charge in Jackson) since a wire is 1-dimensional. Nonetheless, I would expect the oscillations of these lumps along the wire to attenuate over time due to energy radiated by electromagnetic fields (there is a non-trivial motion of charges) and by heat (non-zero resistance along the wire).
