Certainly, it is known that the Lord of the Rings was interrupted for a prolonged spell in 1944-6 (when the Notion Club Papers were drafted) because of difficulty synchronizing the phases of the moon between different parts of the narrative. In fact he never quite managed to solve this problem - but it is a rather obscure matter, and doesn't mean much.
Yet Tolkien made the elementary mistake of recurrently describing his protagonists observing the New Moon Rising at night - when in fact the New Moon rises only during the day - after dawn, following the rising sun - indeed when it is newest the moon is invisible due to being lost in the light from the nearby sun.
(The time to see the New Moon is just after sunset, in the evening - when the New Moon is setting, not rising.)
The first comes from The Hobbit where Bard shoots Smaug the Dragon at the rising of the moon when the moon rose above the eastern shore and silvered [Smaug's] great wings... the waxing moon rose higher and higher.
Then we see something similar in the drafts of The Lord of the Rings published in the History of Middle Earth as The Return of the Shadow when Christopher Tolkien notes:
My father no doubt made this change on account of what he said about the Moon; for there was a waxing moon as the hobbits approached Weathertop, and it was 'nearly half-full' on the night of the attack: the attack was on 5 October...and there could not be a full or nearly full Moon on 24 September, the night passed with the Elves in the Woody End... On that night it must have been almost New Moon. … But it is an odd and uncharacteristic aberration that my father envisaged a New Moon rising late at night in the East.
Nonetheless, this mistake persisted into the published Lord of the Rings - so I conclude that the mistake was not uncharacteristic; but happened because JRR Tolkien believed that the moon could rise at night, after the sun had set; even when the moon was new or 'young' as he describes the rising moon after sunset at Weathertop in the published LotR.
Perhaps it is related to the very origins of Tolkien's Legendarium, when he envisaged (as many people vaguely do) that the sun was created to illuminate the day, and the moon to illuminate the night - albeit going through phases or waxing and waning; so that there is always a light in the sky.
From The Book of Lost Tales Volume 1 - Chapter: The Tale of the Sun and Moon:
...for twelve hours shall the Sunship sail the heavens and leave Valinor, and for twelve shall Silpion's pale bark [i.e. the moon] mount the skies, and there shall be rest for tired eyes and weary hearts.
It is strange, but perhaps not surprising, that such a belief (sun by day, moon by night) is so common given the contrary evidence avaiable to all - and that the crescent moon is mostly a daylight object if visible at all against the bright sky...
But there, perhaps, is the answer. The moon is only noticed by most people when it is near the full, and such a moon is mostly a night sky object - so perhaps they assume (when they bother to think about it) that the moon is only visible at night, rising at dusk and setting at dawn...