Out of curiosity, I searched both of Tolkien's The Hobbit and The Lord of the Rings for "nook":
[The cave] seemed quite a fair size, but not too large and mysterious. It had a dry floor and some comfortable nooks. At one end there was room for the ponies; and there they stood (mighty glad of the change) steaming, and champing in their nosebags.
The Hobbit, Chapter 4: Over Hill And Under Hill, p.53
A little later Frodo came out of the study to see how things were going on, and found [Lobelia] still about the place, investigating nooks and corners, and tapping the floors.
The Fellowship of the Rings, Chapter 1: A Long-Expected Party, p. 57
The westering sun was caught into clouds, and night came swiftly. They slept as well as they could for the cold, turn and turn about, in a nook among great jagged pinnacles of weathered rock; at least they were sheltered from the easterly wind.
The Two Towers, Book Two, Chapter 1: The Taming of Sméagol, p. 227
‘We had better try a way back southwards along the line of the cliff, I
think,’ said Sam. ‘We might find some nook there, or even a cave or something.’ [...] They did not find the going any easier at the broken feet of the Emyn Muil. Nor did Sam find any nook or hollow to shelter in: only bare stony slopes frowned over by the cliff, which now rose again, higher and more sheer as they went back.
ibid., p. 237
Interestingly, nook and cranny doesn't appear at all. As to whether Tolkien's use of nook is "idiomatic", "appropriate" or "common", I'm not sure. I can say however that you'd be in good company.