It's not exactly clear what you mean by "velocity" in that question. Drift velocity? Fermi velocity? (and others) These two differ by multiple orders of magnitude, so it's easy to wrongly conclude that another physicist doesn't have a clue.
Intuitive explanation: drift velocity is the overall streaming velocity of the moving electrons, fermi velocity is the velocity of a single electron. The reason these are different is that electrons move back and forth heavily, thus the fermi velocity is high. But under normal voltages, they move just slighly more in one direction than the other on average, which is the drift velocity. Of course this is highly simplified, and the real story is quantum mechanical.
maybe i'm being over-sensitive here, but it really feels like you're making excuses and trying to hide behind details. surely it's obvious i mean drift velocity from the context (how can you call fermi velocity classical?).
i don't want to drop names, or pull rank, or argue from authority, but this comment is based on memories of a happy afternoon chatting with other students. none of them said "oh, i don't understand, do you mean drift or fermi or one of the many other velocities i can think of?" instead, to a man or woman they said some random large number than stared, then did the maths, and then burst out laughing.
they were smart people. and i admit i was one of the dumbest (and i didn't come up with the question - i can't remember who did). and none of them felt the need to make excuses or smokescreens about learning something new.
I didn't read the "(a purely classical number, but most won't have a clue)" as being part of the question, but rather as commentary about people answering the question. Also "how fast do electrons move" does really suggest the velocity of an electron and not the net average velocity...anyway, calculating drift velocity is easy. From the current and the charge of an electron, calculate how many electrons are passing through the wire per second. Then calculate the number of electrons in the wire from the material properties. The answer is the ratio between these two, multiplied by the length of the wire. Since this contains many numbers that most people (even physicists) don't know off the top of their head (density & weight of copper, avogadro's number, electron charge) this is a bit hard to guesstimate within an order of magnitude without looking things up, but it will be a small number of meters per second.
Intuitive explanation: drift velocity is the overall streaming velocity of the moving electrons, fermi velocity is the velocity of a single electron. The reason these are different is that electrons move back and forth heavily, thus the fermi velocity is high. But under normal voltages, they move just slighly more in one direction than the other on average, which is the drift velocity. Of course this is highly simplified, and the real story is quantum mechanical.