Thanks to Dunsany scholar Martin Andersson -- who wrote an interesting piece on Lord Dunsany and the Nobel in 2018* -- we now know that Tolkien was nominated for the Nobel Prize in Literature at least three times:
in 1961, when he was nominated by C. S. Lewis.**
in 1967, when his name appeared on the (alphabetical) long list as #58 of 70 nominees.
and in 1969, when he was #90 on the long list of 103 names.
So far as I know he did not make the short list any of these times.
This was not Tolkien's first encounter with the Nobel prize. Back in 1954 he had served as a nominator rather than nominee, putting forward E. M. Foster, that quintessentially English author, for the honor. Tolkien was clearly chosen for his position as Merton Professor of English, and it's interesting to note that his nomination of Forster was seconded by Sir David Cecil*** and thirded by F. P. Wilson, all three professors of English at Oxford.
We know that C. S. Lewis put Tolkien's name forward in 1961, no doubt from his status as professor of Renaissance literature at Cambridge.
The nominator in 1969 was R. E. Wycherley, an archeologist and professor of Greek ("grekiska") at Univ. of North Wales in Bangor.
Of his fellow nominees on the 1969 list, twelve did go on to win the prize:
Samuel Beckett (that same year, 1969),
Alexander Solzhenitsyn (1970),
Pablo Neruda (1971),
Heinrich Boll (1972),
Patrick White (1973),
Eyvind Johnson (1974),
Harry Martinson (also 1974),
Eugenio Montale (1975)
Elias Canetti (1981),
Jaroslaf Seifert (1984)
Claude Simon (1985)
and Gunter Grass (though he had to wait for it thirty years, 1999).
Of these, I have to confess that I've read only two: a lot of Beckett (all his plays and even a few of his novels) and one by Solzhenitsyn (his Nobel Prize speech, which we were required to teach to college Freshmen at Marquette).****
Tolkien doesn't have to worry on one account: the Nobel committee is famous, with the hindsight of history, for passing over many of the greats -- such as from the 1969 list not just Tolkien and Forster but also Auden, Frost, Nabokov, Larkin, and, notoriously, Borges.
*appeariing in THE GREEN BOOK, vol. 11, 2018.
**the following year CSL nominated Rbt Frost -- an excellent choice and testimony of how highly he rather the New Engander's work
***fellow Inkling, distinguished biographer, and bete noir of F. R. Leavis
****which the students didn't much care for, though at least Solzhenitsyn fared better than Chesterton though perhaps not as well as Bronowki.