You may wonder how a DC signal can be a wave. Without getting into the math, a DC signal can be decomposed into into an infinite sum of waves (Fourier analysis).
Pedant speaks:
Well, you're talking about a step function, strictly speaking, when you turn on a DC signal, so it's not a "DC" signal, which will have support only in the DC basis vector, but rather a step function, which has support at all frequencies, with 1/f amplitude, if I recall correctly. (it's an integrated impulse, which is flat, so it's 1/f, yes...)
If you're asking about the movement of individual charges then that depends on drift velocity which is determined by the properties of the material at the atomic scale.
Some numbers to consider. 1 mole of electrons is 6.02 * 10^23. That's about the number of electrons in 2.2 ounces (give or take) of copper. That's a few feet of not really big sized wire.
1 amp is 10^19 electrons/second passing a given point, give or take. So, 6*10^4 amps is what you get if one mole of electrons passes a point in one second.
That is a LOT of current. If you put 1 amp through the wire (say it's 10' of wire, which would be moderately small for such a current at 1 mole of copper), you'll have the electrons (we're assuming DC here) drift 10'/(6*10^4) feet in one second.
But the signal will get there with most of the speed of light. Typical propagation is well over .5 C, most wires are about .8 C, and the slowest stuff I'm aware of is about .3 C, and that's not what we use for electrical wiring (and is kinda rare, and used for delay lines).
Figure .8 feet per nanosecond, as a good rule of thumb.