Above him was a black starry sky. Suddenly a pale light appeared over the crown of Weathertop behind him. The waxing moon was climbing slowly above the hill that overshadowed them, and the stars above the hill-top faded. ... ‘Look!’ said Merry. ‘The Moon is rising: it must be getting late.’However, a waxing moon rises during the daylight hours, not at night!
Oddly enough, just a few pages earlier, while describing events taking place two nights previously, Tolkein mentioned the phase of moon accurately. “The moon was waxing, and in the early night-hours a cold grey light lay on the land.” That is correct: a waxing moon rises during the daylight hours and in the early night-hours it is still above the horizon.
Here's a speculative attempt to figure out in some detail the state of the moon the day of the incident at Weathertop - October 6 S.R. 1418. In appendix D, it says the our New Year's Day corresponds more or less to the Shire January 9. That means that the full moon - which they saw on January 8th S.R. 1419 when the Company reached Hollin - would correspond to our December 31st. There was a full moon in London on Dec 31, 1933, which can be used as a reference point. https://www.timeanddate.com/moon/phases/uk/london?year=1933
Also our 26th of September is probably a good approximation for October 6 S.R. - the date of the incident at Weathertop.
On September 26, 1933 the moon was half full and waxing https://www.timeanddate.com/moon/uk/london?month=9&year=1933.
6:52 sunrise 26th September
6:54 sunrise 27th September, the next morning
16:23 moonrise the next afternoon
Daylight savings time was in effect during September in 1933 in England - not sure about Middle Earth on the corresponding date - but the intervals between sunrise/sunset/moonrise/moonset would not be effected.
So by that rationale, the moon would have never been above the horizon during the night of the incident at Weathertop.