This is one of those questions that gives most EEs a bit of headache because it is rather difficult to get one's brain wrapped around what is theoretically going on with electromagnetic waves. The truth of the matter is it is not quite as simple as we first imagine.
Here is how I rationalise it.
First and foremost you need to separate the notion that electrons have anything directly to do with electromagnetic waves. They don't. EM waves propagate without the need for any material. In a vacuum they propagate at the speed of light, when the wave encounters a material they are slowed by the materials physical properties.
Having grasped that bit of wisdom, it is easy enough to understand that when you apply a voltage to one end of a bit of wire, it takes time for you to be able to detect that voltage at the other end of the wire. That voltage propagates down the wire at the speed of light for that conductor material. In effect you created, or more accurately, changed the E-Field at one end and the change wave takes time to get to the other end.
Now consider instead modulating the applied voltage, that is, applying a signal to the wire. If we break up that signal in the time domain into infinitely small periods you can see from what we just discussed that there will be that propagation delay for the instantaneous change at one end to reach the other. The E-Field must "carry" those changes to the other end. Again, it is important to remember, this has nothing to do with electrons.
So, to summarize so far, the electromagnetic waves are the carrier for your signal not the signal itself. EM waves happen to be transverse, and that does not change.
The signal, or whatever voltage wave you are transmitting, is effectively a modulation of that carrier, and is usually longitudinal. You are setting up, or transmitting, a difference in the electrical field. It's those local differences that excites the electrons in your conductor and make them move in reaction.