There are a few issues that regulate the speed of CD:
- Upwelling: of magma at the rift zones.
- Inertia: the plates undergoing CD are big and heavy.
- Firmness: of the asthenosphere (upper mantle). That's what the continents float on, and it's movement causes a plate to move. Controlled by heat in the lower mantle and outer core.
- Friction: of rock at the subduction zones.
Thus, to speed things up...
- Hotter core to make more magma rise at the rift zones.
- Lighter continents?
- The hotter core will also soften the asthenosphere, making it move faster.
- Slipperier rocks (like talc) at the subduction zones.
The tidal flexing caused by a massive orbital satellite ultimately produces frictional heating. This topic is covered here:
Jupiter's moon Io is an example: this moon is largely heated by tidal forces. So too your earth with faster continental drift - a new satellite might impart tidal forces such that the mantle heats up and drift speeds up./
2: Interior heats up more - either because of production of more rapidly decaying daughter radionuclides or the addition of fresh radionuclides.
Another way to heat the interior and move plates faster would be to have the nuclides heating the earth produce more heat. This could happen via an increase in decay products to isotopes which more rapidly decay and so over the short term produce more heat. Heating of the earths interior is caused by radioactive decay.
In some circumstances, decay of certain isotopes produce daughter isotopes which decay faster and so more heat is produced. I asked about this here.
Here is a diagram from @Farcher's answer. The third instance, non equilibrium shows a place in the course of decay where the heat transiently increases.
You could also just add more fuel to the fire. For example, suppose a collision event entailing a meterorite comprised largely of uranium, or americium or the like. The dense meteorite would make its way towards the center of the earth and once there contribute to heating of the core.
If you are not wanting to speed up plate tectonics but just have it be faster from the get go, have your planet with more radioactive nuclides in the core from the start.