You’re thinking of

Knee-Deep in Thunder
by Sheila Moon (Macmillan, 1967; very widely re-issued by Scholastic in the 1970s-1980s).

Why it matches the points you remember

• Three children from different backgrounds are the protagonists.
– Maris (a girl from California),
– Hal (an English boy) and
– Raf (a boy whose father is from India)
are swept out of the ordinary world and magically reduced to insect size in the roots and leaf-litter of a redwood forest, which for them has become an underground fantasy kingdom.

• A giant “inchworm”.
The children travel for a long stretch astride a huge green looper (another name for an inch-worm). Saddles are strapped on at every second segment, so the middle rider rises and falls the most. When guards pursue them through a narrow root-tunnel the looper arches its back; the highest saddle is rammed against the roof, its rider is knocked off and separated from the others – exactly the incident you describe.

• Glowing mushrooms.
Deeper in the tunnels Raf is so hungry that he eats some of the kingdom’s phosphorescent toadstools; the fungus makes his skin and clothes shine in the dark and, when he blunders into the enemy camp, the soldiers take him for a ghost.

• Evil ruler.
The land is being terrorised by the usurper Prince Kuen; the children join the rebel insects and other creatures to overthrow him.

Because the book was sold for years through the Scholastic classroom catalogues it turned up in a great many school and public libraries, which is why so many people who were children in the 1970s–1980s remember it.