For conduction in copper wire:
The aspect of current that is _net_ flow of charge is _fast_ (moving at a large fraction of the speed of light). If you have a loop of conductor many miles long, and you apply a potential at the near end terminals, and measure the current flowing at the near end terminals, you will measure the same current flow at the far end of the loop, with a delay that is of the same magnitude as it would take light to reach the far end measurement location.
Any measurement or definition of current that depends upon the _net_ flow of charge will demonstrate that current moves very quickly.
The aspect of current that depends upon the rate of movement of specific charge carriers is _slow_, in fact very slow.
Any measurement which depends upon the rate of charge carrier flow, it will measure a very low velocity.
An example of such a measurement is 'hall voltage', the voltage induced across a conductor in a transverse magnetic field. When a charge carrier moves in a magnetic field, it is pushed in a direction that is perpendicular to both the direction of motion and the magnetic field. This produces a voltage across the conductor, which can be measured.
http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/hall.html
I generally hate the electricity as water flow analogies, but consider this: given a pipe with water flowing through it at a rate of 1000 gallons per minute, would you say that the water is flowing quickly or slowly? How about if I tell you that the pipe is 1" in diameter? How about 200" in diameter?
I don't see how you can say current is fast or slow without explicitly defining what aspect of charge movement you are calling current. If you use the SI definition of the Ampere, you are using _net_ flow, which is clearly fast. If you are using the actual movement of charges, then it is clearly slow.
-Jon