Although these books do seem to fall somewhere in the crack between outright infringement and fair usage David Day seems to be relying on a number of legal figleaves to protect himself:
The books always prominently marked as being written by David Day rather than by J. R. R. Tolkien
The books contain a disclaimer on the first page that these books are not licensed works, nor works directly written by Tolkien
This book has not been prepared, authorized, licensed, or endorsed by J. R. R. Tolkien’s heirs or estate, nor by any of the publishers or distributors of the book The Lord of the Rings or any other work written by J. R. R. Tolkien, nor anyone involved in the creation, production or distribution of the films based on the book.
The books contain a disclaimer in the marketing blurb for each book highlighting that these are derivative works, not originals.
This work is unofficial and is not authorized by the Tolkien Estate or HarperCollins Publishers.
The books each contain a preface extensively describing the "transformative" aspects (e.g. what input the author has provided above and beyond simply quoting from the original author).
His An Atlas of Tolkien, for example contains this in the preface:
This Atlas is a compilation of specially commissioned and created art
from some of the most talented fantasy artists of the past four
decades. It began with the enormous investment in original full-colour
art exclusively commissioned for the publication of A Tolkien Bestiary
(1979) as the first ever fully-illustrated reference work on JRR
Tolkien. Subsequently, new original artwork was created for Tolkien:
the Illustrated Encyclopedia (1992) and for The World of Tolkien: the
Mythological Sources of Lord of the Rings (2002).
His A Dictionary of Tolkien has this:
The Tolkien Companion was written in celebration of this aspect of J. R. R. Tolkien’s genius. It was compiled and designed as a compact and easy-to-use guide to Tolkien’s world. The purpose is to inform and entertain those readers who wish to use the Companion to help them in their personal exploration of the extraordinarily complex invented world and mythology of Middle-earth and the Undying Lands.
No-one reading those could be in any doubt that these are not original writings by Tolkien and that the contents include additional work/s not by Tolkien.
Day also posits the argument that his works enhance the value of Tolkien's works rather than diminishing it.
'Their father's work is a sacred text to them and they feel that
anything that doesn't come from them, the Tolkiens, can only ruin
it'... [Day] laughs at the furious letter he received from Christopher
Tolkien calling him 'an ass' and 'more like a burglar than a writer
... In the past two months alone, stores have sold more copies of The
Lord of the Rings than the typical annual totals before JRR's death."
Daily Express - The Sad Legacy of Tolkien's Fable - Google Groups Reproduction
It's also fairly telling that Chris Tolkien personally wrote him an unfriendly letter. You don't badmouth someone if you think there's any possibility that you can successfully sue them instead.
So why don't they sue him anyway?
There's every possibility (and in fact every likelihood) that they'd lose in fairly short order. Day would undoubtedly mount a robust "fair use" defence under at least two of the four elements that need to be considered when determining if an item is copyright infringing; that his books are substantially transformative (by adding entirely new content and elements of commentary) and that they're not harming sales of Tolkien's original works (since there are no other comparable works published by the Tolkien Estate and since, almost by definition, someone would have to already own an official Tolkien book before they'd consider buying a book of commentary about it).
In addition to being an enormous waste of money and time, and coincidentally providing Day with hundreds of thousands of pounds of free publicity for his books, losing such a case could harm the Estate's future ability to fight against more egregious infringements.
Given the age of these works and the fact that their rights haven't always been continually asserted, it's at least possible that an unwelcome judgement might open the floodgates to other derivative works or, in an apocalyptic worst-case scenario, the judge might even rule that elements of the original copyright no longer apply. The risk probably just isn't worth it to swat this fly.