Discussion on the drude theory of metal usually begin with the case of zero magnetic field so that the force acting on the electrons is just the one from the electric field. But then, this electric field is related to a charge current density. How is it possible to have a current but there is no magnetic field?
The Drude model is very crude, but it results in the current density J being proportional to electric field E, which in turn brings us to the current I being proportional to voltage V (V = RI). So it provides an reasonable explanation for electrical engineering's most important equation.
Once you have J, you can extract I, and calculate the magnetic field as usual (Biot-Savart Law). It's neglected at first, because the model provides useful results regardless, but you can always consider it if need be.