Thanks to Kevin McCall to tipping me off to the existence of this children's story, published in 1954 -- just before the West's discovery of psilocybin mushrooms, after which the title would have had rather different connotations!
The titular mushroom planet, called Basidium X, is extremely small, and is not actually a planet in the modern sense but a satellite of Earth, closer to us than the Moon, but invisible to the naked eye and to ordinary telescopes. It is discovered by Mr. Tyco (!) Bass, who lives on Earth but is descended from mushroom people, by putting a special filter on his telescope involving a stroboscope and polarized light. Having a telepathic sense that his people live on that planet and are in distress, he recruits two human boys to build a spaceship and fly there to save them.
When the boys arrive on Basidium, they deal primarily with two mushroom people who bear the title Wise Men. The boys have brought a hen with them on their voyage, which turns out to be the key to saving the mushroom people, who are suffering from a sulfur deficiency which can be rectified by eating egg yolks. Having never seen eggs before, the mushroom people call them "magic stones."
Parallels to Alizio and the Little Skinny Planet are extensive. I have discussed evidence that the Little Skinny Planet is a satellite of Earth. The mushroom people are short and bald, like Tim and Patrick, and are extremely thin. Tyco Bass is described as having "thin spindling arms" -- cf. the "spindle legs" of the giraffes on the Little Skinny Planet. Recall that those giraffes were optically unusual -- shimmering with impossible colors -- and that Basidium's invisibility to ordinary telescopes implies something similar.
The little bald men Tim and Patrick correspond to the two blue-clothed wizards in the Joseph story, who it is suggested are the biblical Wise Men even though there are only two of them. Mebe and Oru, the two Wise Men in The Wonderful Flight to the Mushroom Planet, are little bald men. In the Alizio story, Patrick eats all of Alizio's Hidden Treasures. In The Wonderful Flight, Mebe and Oru are the first to eat egg yolks and be restored to health. In one of Bilbo's riddles in The Hobbit, egg yolk is called a hidden treasure:
A box without hinges, key, or lid,Yet golden treasure inside is hid.
Of course William Wright will appreciate the "magic stone" angle. (Note also that the author's name is Eleanor.)
One other possible link: Prior to my Tim dreams, my only experience dreaming in Latin had been under the influence of nutmeg, when I had a conversation in that language with a mantis shrimp. Mantis shrimps are famous for their ability to see many more colors than we can, and to discern different kinds of polarized light.