Diana Glyer is a treasure. I just listened to another interview with her. I read and loved Bandersnatch, and just put The Company They Keep, which is the expanded version, on my wishlist. The prevailing opinion among both Lewis and Tolkien scholars for decades was that though they met together once and sometimes twice weekly for thirty years to discuss their own writing, along with other topics, they did not influence each other much. Viewed in that light, it seems unlikely. This is because they both said as much, and because literary criticism tends to define "influence" from afar, so that only clear imitation is visible. Glyer goes much deeper, showing that many, many things, from encouragement/discouragement, overall world view, and acceptable choice of topics are all influences, yet she also recounts influences of clear influence once one compares the diaries of all to the drafts of works in progress. For example, after Tolkien read one section of his New Hobbit, several Inklings thought he should use less dialogue and more narration. Tolkien scoffed, claiming they just didn't know what they were talking about. But his next draft had 35% less dialogue and 25% more narration than the previous version. That is the kind of "influence" that is not visible when asking if Dickens influenced Melville.
All just so much fun, interesting, but not so moving. Here's one that caused me to stop my walk, play this piece of the podcast over again twice, and then turn it off to think about it for the next couple of miles. Tolkien was writing his New Hobbit at the request of his publisher, and had gotten through the first few chapters, up to the point where the hobbits had left Hobbiton, had stopped in a quiet place to rest against the roots of a tree, and heard hoof-beats. It was Gandalf arriving on a white horse. Tolkien complained to Lewis that he felt he couldn't go forward anymore, as it all bored even him. He couldn't imagine a reader being interested. Lewis thought he saw the problem. "Hobbits are only interesting in un-hobbit-like situations." Tolkien apparently did not reject this idea out of hand, as he usually responded to advice. He went back that very night and thought that a black rider on a black horse would be more interesting, and more in keeping with the whole of Middle-Earth. He started his rewrite from the very beginning, and by the time he got up to that part of the story again he knew that the black rider was not merely shadowy but a Nazgul, an empty, unearthly Black Rider, and also had some idea why Gandalf might be delayed. After that, he told Lewis within a few weeks, the story was writing itself and he could not keep up. (Tolkien was a binge writer who would do nothing for weeks, then deprive himself of sleep to push through a particular section in a few days.)
So..."influence." What does it mean? Lewis didn't suggest anything about a Black Rider, or deepening the story, nor continuing it further. That was all Ronald, who could legitimately claim (if he chose, though that was not his style) that he had done it all and Lewis done very little. But that was the whole of LOTR right there. The change to a black rider was everything.
BTW, other Inklings have also credited Christopher Tolkien's making maps of Middle-Earth not long after this as also having a major influence. JRRT in some ways wrote for an audience of one, Christopher, and seeing him expand the universe, unintentionally tying it in to the huge events of the legendarium of Middle-Earth, likely allowed Tolkien to see/invent the reality that these events were not a small hobbit adventure, but the crux of the main conflict of the many ages of Middle-Earth.
But at this early point, with the hobbits still in the Shire, it was still a "There and Back Again" story, which he expected to end at Frodo's arrival in Rivendell, with perhaps a short section about his return. Fans will notice that Book One ends at the arrival in Rivendell. The next place that Tolkien got stuck was in Bree, where the hobbits arrived, though in precarious danger, and encountered a suspicious-looking "Ranger."
A Ranger named Trotter, who was a hobbit. The year was 1941, and even had the manuscript been complete it could not have been printed because of the paper shortage. So the story sat for a while.