“So what do the Christians do?”
Hmmm... Well the deified rabbi worshiped by Christians celebrated Jewish holidays. If Christians really wanted to follow the example of their rabbi they’d probably celebrate Sukkot (Festival of Booths), Chanukah (this year it’ll be Thanksgivikah because it and T-day run together; next one in 79,000 years.), and at least all of the other major ones, but probably not the minors.
Chanukkah is a minor holiday. It's a big deal in America because it usually occurs somewhere around Xmas time and because the Chanukkah story has unfortunately been torn from its original Theocratic context into an alien one where Antiochus is turned into a "bigot" and the Chashmonayim into a sort of American Civil Liberties Union.
Did anyone notice the major Jewish holidays last month? Very few. Does Purim ever upstage St. Paddy's Day? No (Purim is actually the same class festival as Chanukkah but you never hear about it). And of course Shavu`ot is unknown to practically everyone.
The "major Jewish holiday of Chanukkah" is strictly an American phenomenon.
“So what do the Christians do?”