Wood Wide Web: Already a term in biology. "Research has shown that beneath every forest and wood there is a complex underground web of roots, fungi and bacteria helping to connect trees and plants to one another. This subterranean social network, nearly 500 million years old, has become known as the "wood wide web"." (BBC)
Fungiverse: Most similar to the term fediverse though I never understood the "universe" part of the term. What does it mean in this context? Its more a social network, right? Much more like the internet it is based on a certain protocol.
Fungal/Fungi Web: Shorter than Wood Wide Web and maybe easier to say. In contrast to Wood Wide Web, maybe it's also better to not confuse tech and nature here. I also in general like the term "social web" more, because it emphasizes that it is basically going on top of the usual web just through a new protocol.
I think Wood Wide Web would be best, because it could emphasize that it should be energy-efficient and have the goal of connecting people to collaborate toward a sustainable future. Also: in a story that plays in a world in which humanoid plants live, it would just make sense that they discover the Wood Wide Web at some point. What do you think?
Hehe, that's why I recommended it, it sounds fancy :P.