Readers who have an interest in Arthurian literature should find this interesting for its exposition of Tolkien's source choices. Those who are only interested in Middle Earth, may have trouble associating this book with the Tolkien they know. Christopher provides some help in bridging the gap. Those who are expecting a full-fledged Arthurian experience will be disappointed.
Most of the English speaking world knows of Arthur through Sir Thomas Mallory's 15th century version of the stories. With few exceptions, what appears in the popular media is based on Mallory. The exceptions generally ignore the vast earlier base of Arthurian literature, borrow a few names and incidents, and invent new relationships between the characters and create new narrative. The film King Arthur (2004) is a good example of this.
Tolkien made a conscious choice to focus on the most "English" aspects of the legends.
Arthurian literature before the 12th century would fit on part of one page. Geoffrey of Monmouth sparked interest in the Arthurian stories, starting around 1150, when Arthur was included in his History of the Kings of Britain. Monmouth gave us about 33 pages of Arthurian "history". This was followed by an avalanche of writing in French and German that lasted 100 years, until around 1250. The English versions of the stories first appeared 100 years later, in 1350. One of these was the West Midlands Sir Gawain and the Green Knight, translated by Tolkien and Gordon in 1925 while they were professors at Leeds. The other was the Stanzaic Morte D'Arthure. Gawain and the Stanzaic were used as sources for the Alliterative Morte D'Arthure around 1400. The Stanzaic and Alliterative were sources for Mallory. Gawain is borrowed from Briciu's Feast, an episode in Irish mythology, and adapted to the Arthurian legends.
The importance of this is that Tolkien took the most direct "English" path to Monmouth when choosing his sources. As Christopher states in the comments accompanying the poem, Tolkien used the vein starting with Monmouth, then to the Alliterative, finally to Mallory. This is as close as he could get to an "English" version. Monmouth was born in England, of Breton parents. Mallory was also influenced somewhat by continental versions of Chretien a Troyes and the Post Vulgate, but Tolkien seems to have expanded on Mallory's choice to ignore important aspects of the post-Monmouth continental versions, like the role of Lancelot. He seems to have been interested in purging the continental influences not already present in Monmouth.
It may surprise some that Tolkien, a scholar of language and mythology, once wrote (1951) that England "had no stories of its own..., not of the quality I sought". In the same paragraph he notes the Arthurian legends are "imperfectly naturalized, associated with Britain, but not with English". The Lord of the Rings and its accompanying literature were his attempt to create a mythology for England. It was published starting in 1954.
Tolkien's first attempt to write his own mythology started in 1914. A 28 page "Sketch of the Mythology" was written in 1927. Tolkien started The Fall of Arthur sometime before 1933 and it was abandoned by 1934. He never returned to it. In 1937, he submitted an early version of what became the Silmarillion to the publisher of The Hobbit. The timing of The Fall of Arthur seems to indicate a fleeting hope that he could convert Arthurian literature into a myth for England. However, it is impossible to ignore the many ties this body of literature has to the continent, especially France. Connections to the continent even appear in his brief start, which includes Frisians, and for which the bulk of the text is concerned with Arthur's trip to the continent, leaving Mordred in charge, and Arthur's return from France. Lancelot is French. Many stories in the wider body of the French and German stories are centered on what is now France, especially Brittany. Echoes of this even appear in The Lord of the Rings. "Rohan", for example is a place in Brittany where the plateau meets the rougher ground of Brittany. "Mirkwood Forest" seems to be patterned after the Forest of Broceliande, in Brittany, which is connected to many Arthurian legends, especially those of Merlin, Palamedes, and others.
If the story had been completed, it would attract a larger audience. As it is, it is rather specialized. Those of us in that audience, are very grateful for it.