I think you're right Allyssa that Narnia was published first. But Lewis also read many drafts of LotR and new much about Tolkien's world. From reading Tolkien's letters, I think Tolkien suspected Lewis of imitating him with some of his works (I don't know if these works included Narnia or not). For example, Lewis used the name Numenor, but having only heard Tolkien say it he spelt it 'Numinor' (or something like that). Does anyone know where this word is found?
The word is one of the three science fiction novels Lewis wrote. I think it was the second one "That Hideous Strength", but I'm not certain.
LOTR was finished by 1949, before Lewis had started Narnia, but it took JRRT 5 years to get LOTR published. (This was partly his fault, as he was insisting that publishers put out The Silmarilien at the same time.) Lewis was very familiar with LOTR by 1949. Almost the entire tale was read to the Inklings as JRRT composed it, and Tolkien always gave Lewis great credit for the encouragement Lewis gave him (even to the extent of doubting he would have finished without Lewis' moral support. Not specifics- Lewis apparently didn't make many recommendations; just more along the lines of, "This is fantastic; you must finish." So the time frame makes it clear that the influence goes more from JRRT to Lewis than the other way around. Still, they had been friends for 20 years, and cross currents are possible. Their relationship was quite complex, and they drifted apart in the last ten years of Lewis' life (JRRT isn't even mentioned in the movie Shadowlands, which is about Lewis' marriage toward the end of his life.) I could go on in this vein for awhile, as the two of them had many connections. Probably a good book could be written on the influence each had on the other.