From a Guardian article last year.
The attitude of authors to fanfic based on their works varies immensely: George R.R. Martin disapproves, but generally doesn't make too much of a fuss as long as fanfic is not sent to him. J.K. Rowling is somewhat more supportive of the concept. Some authors are a lot more enthusiastic and even host fanfic on their website, although this becomes legally dubious if the author later makes story decisions in the "official" material that echoes the fanfic.
In the case of The Chronicles of Narnia, the situation is both less and more clear: C.S. Lewis died in 1963, so is not around to make any judgements of his own on the matter. The Lewis Estate has resisted anyone writing "official" new Narnia material, but has been happy to authorise various adaptations for television and film, with Netflix being the latest studio to pick up the rights and begin development work of a new version of the story. Curiously, around the time the first Narnia feature film was released in 2005, the Estate did suggest that new books would be released as well, but nothing ever materialised.
The fanfic in question is called The Stone Table and takes place between the events of The Magician's Nephew and The Lion, the Witch and the Wardrobe, explaining what happened to the rulers of Narnia and the castle of Cair Paravel prior to the events of the latter novel. This is a fertile area for exploration - and the title and premise feel potentially like Lewis inventions - and indeed this period has been discussed many times by Narnia fans over the years.
Francis Spufford - who has previously published exactly one novel (Golden Hill, in 2016, which won the Costa Book Award) - wrote the fanfic for his daughter, who had complained that her father had not written any books she felt like she could read (Spufford has mostly written non-fiction on adult subjects). Spufford spent three years working on the fanfic, which is apparently novel-length, which is not an inconsiderable time period for one fic.
Oddly, Spufford has declined to post the work through any of the established fanfic communities, instead only releasing the first couple of chapters at the urging of friends who have read the full book. Distinguished SF author Adam Roberts has acclaimed the book as being excellent and a worthy addition to the Narnia canon, which is a nice compliment. However, given that the only person who can make that determination is near sixty years dead, it's not particularly germane.
Spufford has apparently now reached out to the Lewis Estate to see if the book can see print in some fashion with their approval. If not, he'll apparently wait the fifteen years until the books leave copyright and publish then, which feels like an odd choice to make when there are plenty of fanfic communities who would be happy to release the book and critique it immediately, for free.
Or, of course, he could follow in the footsteps of the great fanfic trailblazer and innovator E.L. James, change a few names to create a new setting and cash in straight away.