There are only 3 ways to "clean up" the Holy Mysteries: consuming them, washing them out (and pouring the water into the grass) or burning them. When they get onto the floor, we clean them up with communion cloths (which, depending on their type, are then either burned or washed); if you have carpet, then that section should be removed and burned. If it's a portable rug, you may be able to wash it out (again, with the spent wash water / Holy Communion going to the ground, not down the drain). Before they're burned or washed, the priest or deacon will consume any of the Holy Body.
The priest or deacon will be the person primarily responsible for clean-up; it's not something to be "farmed out" to someone else.
There are 3 basic principles to remember in the moment:
1. Don't panic (the world is not ending)
2. Act quickly
3. Be thorough