It's a meaningless question, because all electrons are identical. You could just as well ask how long it would take the electrons to be replaced from the wire into my body; the answer is the same: it doesn't matter!
If you change the problem so that identification is possible, then the answer is either the drift velocity, or the carrier diffusion velocity, whichever is higher. (Few conductors operate with a drift velocity higher than the thermal velocity, i.e., ballistic rather than drift motion. An example is electrons in a Gunn diode, which is really a monode: just a lump of doped GaAs that exhibits negative resistance, thanks entirely to its material properties, no junction needed!)
A possible substitute could be an ionic conductor, like heated ZrO
2, which carries current through mobile O
2- ions and lattice defects.
The experimental design might be like so:
Set up a ZrO
2 plate, with metallized grids on either side, with an atmosphere of
18O
2 on one side. The exchange of O
2 with the ceramic surface, introduces different nuclear isotopes into the composition (which starts as mostly
16O). As
16O is displaced with
18O isotopes, labeling is achieved, and observation of the process is possible.
You might analyze the results by drawing various currents through the samples, for different times, at different temperatures, then sectioning the samples. (The samples should be relatively stable at room temperature, because the ions freeze in place.) An atomic fluorescence spectrometer should be able to resolve the isotopic differences without too much hardship, I think?
It would be an interesting graduate experiment, but rather useless in results, as it should be expected to give the identical result that chemical diffusion and electronic transport are known to achieve. (And, anomalous results could depend on impurities in the materials, so you might not learn anything even from very precise measurements.)
Tim