For those crashes which don’t lead to instant incineration, this job is carried out by the fairy squad. They don’t like it but as Hanastrew says “there is nothing quite like it for getting you used to the sight of blood. The clever bit is to ensure the blood is not yours.”
Instant incineration tends to happen when the accident occurs at great speeds or when the broomstick owner has a death wish and has cursed their vehicle to explode on impact with anyone else’s (some of the older witches see this as an honourable way to exit the Fairy Kingdom).
When it comes to clearing up after accidents where flying took place using boot power (mainly the sprites), these tend not to be fatal (given the limited power of even the best quality boots) but the fairy squad inevitably will be breaking up the fighting taking place between the sprites in the accident.