When the voltage is increased does the speed of electrons increase or does the electron density increase?
In a conductive material such as a metal, for all practical purposes, current depends only on the speed of the electrons. The electron density does not change because each metal atom has already given up all of its valence electrons; releasing further electrons would require a very large energy input.
In an insulator or semiconductor, the density of charge carriers may increase during electrical breakdown. This occurs in avalanche diodes, neon lights, lightning bolts, and elsewhere.