I know people that work in mycology and plant pathology. My impression is that the "wood wide web" is generally a well-accepted and well-established concept. It's commonly accepted that mycorrhizal networks exist, that they benefit both plant and fungus, and that they allow propagation of resources between trees. There is some debate over the anthropomorphisation and the idea of "mother trees", but that's kind of what you expect when it comes to science popularisation in the media.

FWIW the book "Entangled Life" is okay as a popular science book, but Sheldrake isn't really a practicing research mycologist, so you can't expect too much from it scientifically. And it has some... slightly weird bits (do you really want to picture the author rolling around naked in some leaves?).