Tolkien addresses the use of some of his words by C.S. lewis in a letter to Dick Plotz on September 12, 1965. The letter is numbered 276 in The Letters of J.R.R. Tolkien.
This is the relevant part of the letter:
Núminor. C. S. Lewis was one of the only three persons who have so far read all or a considerable pan of my 'mythology' of the First and Second Ages, which had already been in the main lines constructed before we met. He had the peculiarity that he liked to be read to. All that he knew of my 'matter' was what his capacious but not infallible memory retained from my reading to him as sole audience. His spelling numinor is a hearing error, aided, no doubt, by his association of the name with Latin nūmen, nūmina, and the adjective 'numinous'. Unfortunate, since the name has no such connexions, and has no reference to 'divinity' or sense of its presence. It is a construction from the Eldarin base √NDU 'below, down; descend'; Q. núme 'going down, occident'; númen 'the
direction or region of the sunset' +nóre 'land' as an inhabited area. I have often used Westernesse as a translation. This is derived from rare Middle English Westernesse (known to me only in MS. C of King Horn) where the meaning is vague, but may be taken to mean 'Western lands' as distinct from the East inhabited by the Paynim and Saracens. Lewis took no pan in 'research into Númenor'. N. is my personal alteration of the Atlantis myth and/or tradition, and accommodation of it to my general mythology. Of all the mythical or 'archetypal' images this is the one most deeply seated in my
imagination, and for many years I had a recurrent Atlantis dream : the stupendous and ineluctable wave advancing from the Sea or over the land, sometimes dark, sometimes green and sunlit.
Lewis was, I think, impressed by 'the Silmarillion and all that', and certainly retained some vague memories of it and of its names in mind. For instance, since he had heard it, before he composed or thought of Out of the Silent Planet, I imagine that Eldil is an echo of the Eldar; in Perelandra 'Tor and Tinidril' are certainly an echo, since Tuor and Idril, parents of Eärendil, are major characters in 'The Fall of Gondolin', the earliest written of the legends of the First Age. But
his own mythology (incipient and never fully realized) was quite different. It was at any rate broken to bits before it became coherent by contact with C. S. Williams and his 'Arthurian' stuff – which happened between Perelandra and That Hideous Strength. A pity, I think. But then I was and
remain wholly unsympathetic to Williams' mind.
This was mentioned in the 2005 paper "The Inklings: Connecting the Unconnected: Genre, intertextuality and intersemiosis" by Tommi Nieminen (Department of General Linguistics, University of Helsinki, Finland):
When Tolkien and Lewis made an mutual agreement to write two science fiction stories—Tolkien of a time travel, Lewis of a space travel—, and Lewis produced the first part of the Ransom trilogy, Out of the Silent Planet (1943), it was just as much an allegory as were his previous works, or the later ones such as the Narnia series (1950–56). Even the names borrowed from Tolkien to the Ransom novels—taken not from Tolkien’s writings but his reading aloud The Silmarillion stories in the Inklings meetings—were given so different second-world interpretations that they do not carry anything of Tolkien’s world to Lewis’s.
This was the only confirmed use that I could identify. Others have theorized that other words were borrowed from Tolkien's conlangs, such as the name of the palace in Narnia, Cair Paravel. "Cair" in Sindarin means "ship". This is discussed in more depth on TheTolkienForum.com, but ultimately it looks like both sources of Cair come from the welsh word "Caer", which means a stronghold.
The only confirmed use I found was the use of names by Lewis in the Ransom trilogy mentioned above.