Well, the most simple approach here is indeed just the Drude model. You start thinking about a conduction electron, which moves through the wire, driven by the applied constant electric field ##E## (in direction of the wire). This electron is free exept for collisions with all kinds of stuff like impurities in the crystal lattice, lattice vibrations "phonons" etc. etc. We don't care for this but just assume that there's a friction force propotional to its velocity, ##F=-\alpha m v##. Thus the equation of motion of this electron reads
$$m \dot{v}=q E-\alpha m v.$$
What happens is that in the beginning when the electric field is switched on the electron accelerates and after a relatively short time reaches a velocity, where the friction just compensates for the electric force, and then you get a constant velocity, the drift velocity. That gives, because then ##\dot{v}=0##,
$$v=\frac{q E}{\alpha m}.$$
Now the total current density is given by ##j=n q v##, and thus you get Drude's formula
$$j=\frac{n q^2 E}{\alpha m}=\sigma E,$$
and thus for the electric conductivity
$$\sigma=\frac{n q^2}{\alpha m}.$$
For the drift velocity you thus find
$$v=\frac{j}{q n}=\frac{I}{A q n},$$
where ##A## is the cross section of the wire. For a realistic estimate for household current, see
https://en.wikipedia.org/wiki/Drift_velocity#Numerical_example
As you see, you get a tiny drift velocity of ##v \simeq 2 \cdot 10^{-5} \text{m}/\text{s}##. The smallness if of course indeed due to the huge amount of conduction electrons in the wire, i.e., because the number density of counduction electrons ##n## is a large number.
In the final sentence of this paragraph at Wikipedia concerning the Fermi velocity they should rather talk about Fermi speed, i.e., the magnitude of this velocity. The reason is what I told already in my previous posting: The electrons' thermal motion (i.e., in this case their Fermi motion) is random, i.e., it averages out to 0 when averaging over quite small volumes and/or times.
