Charles tells Eliza, "I've even heard a rumor of two brothers in North Carolina who are working on a flying machine." Since he disappears the next day, December 18, 1903, he says this on the actual day of the Wright Brothers' first flights.
The three books Charles reads to research time travel are: The Time Machine, A Christmas Carol, and A Connecticut Yankee in King Arthur's Court.
As noted in the Goofs section, it is impossible for a second full moon in December to occur on the 18th of the month or on Christmas Eve. In the original novel by Alexis Stanton, Charles winds the clock on December 18, 1902 (rather than 1903 as in the film) during a "red moon" lunar eclipse, and in the present day another red moon lunar eclipse takes place on Christmas Eve, rather than a second full moon of the month, prompting him to attempt a return to the past.
Ryan Paevey and Nelson Wong appeared together in Noël au palace (2019) and Un nouveau chapitre pour Noël (2018).