What techniques do both authors use to uphold the imaginary without demonstrating the children to be deluded in Swallows and Amazons by Arthur Ransome and The Story of the Treasure Seekers by E. Nesbit? Explain how the imaginative functions as a recuperative possibility (for the readers).

E. Nesbit's novel was published in 1899, thirty-one years before Arthur Ransome’s 1930 book. He is among the “children's authors” who acknowledged Nesbit's influence. One reason both books would likely appeal to children is the amount of agency associated with the children. They are well realized people who do not regard themselves primarily as being steered by adults. The elements that would be considered imaginary are rendered in a straightforward way, and the reader easily adopts the child's perspective. The fantastic is rarely marked as such, so the books seem to offer a more authentic child's-eye view.
Ransome tells the story of the children in two families, who call themselves the Swallows and Amazons of the title. They are actually the Blacketts and Walkers. The children are devoted to pirate tales, obviously influenced by Robert Louis Stevenson's Treasure Island. They turn their camp-outs on an English lake island into another land, Wild Cat Island, peopled with pirates, and engage in a war with Amazons.
Even when discussing practical sailing techniques, the children's fantasy identities are used.

"Isn’t there another ring under the boom, close to the mast?” asked Queen Elizabeth.

In Nesbit’s novel, the six Bastable children set about helping their father recover the family fortune. Oswald Bastable, the self-styled although ostensibly secret protagonist encourages us, through first-person voice, to regard highly the children's achievements. While they persevere and even triumph, they are not the idealized angels that a third-person adult narrator might have presented.
The family badly needs money, the narrator reveals:

The fortunes of the ancient House of Bastable were really fallen.

To get money, children's fantastic ideas seem as viable as realistic ones. They set out to work—child detectives could be as likely as Holmes—and encounter fanciful characters but do not live in a fantasy world.
Noel, for example, told his siblings that

he had not made up his mind whether he would print his poetry in a book and sell it, or find a princess and marry her.

Dora matter-of-factly preferred the title method.

Let's dig for treasure. Not any tiresome divining-rod—but just plain digging. People who dig for treasure always find it. And then we shall be rich.
https://books.google.com/books/about/Swallows_and_Amazons.html?id=3cntt4wcgh0C&printsec=frontcover&source=kp_read_button

https://www.gutenberg.org/files/770/770-h/770-h.htm