What's the fastest legal way for the nutrients in a dead body to re-enter the ecosystem?
The Parsis and other Zoroastrians dispose of their dead in a so-called Tower of Silence, where the body is left on a raised platform to be eaten by scavengers, particularly birds such as vultures.

This is possibly (probably?) the quickest way for nutrients to re-enter the wider ecosystem, as scavengers usually work a whole lot quicker than microbes.

Tibetan tradition also contains a ritual known as the Sky Burial, which achieves the same sort of thing. The Tibetan version may be even quicker, as once the scavengers have stripped the body, the bones are ground into a paste along with flour and yak's milk/butter to be fed to a second round of scavenging birds.