The electrons accelerate in the electric field, but reach an average velocity after about a scattering time. The characteristic scattering times for electrons in a piece of metal are typically very short, say in the neighborhood of 10-14
sec. That means that a current density proportional to electric field gets established very quickly.
In the sort of situation you're asking about, that current then shifts charge so as to reduce the electric field, so the current density falls proportionately. You can actually convert the conductivity of the metal into a characteristic decay rate for the electric field. In fact, in the CGS system of units the conductivity is given directly in inverse seconds, so no conversion is needed in those units. The conversion factor from conventional units is about 1 mho/cm -> 1012
/s. For a typical piece of good conductor, with conductivity of about 105
mho/cm, that would give relaxation times faster even than the short time it takes for the field to set up the current density, a result which isn't physical. In practice, that means that within about one electron scattering time, the little shift in positions of the electrons is enough to make the potential about uniform. Depending on geometry, magnetic fields produced by the current flow can slow down the process. Those magnetic induction effects become very important for applied ac fields.
In case I've still missed the core question, please follow up. Meanwhile, you might look for a copy of Purcell's book on Electricity and Magnetism, which does a great job of conveying a feel for how these things work.
(published on 11/21/11)