The natural choice for Christmas viewing or listening is "The Blue Carbuncle." My personal favorite rendering is the Granada Television offering starring Brett & Burke (circa 1983). The various radio dramatizations of "The Blue Carbuncle" are not entirely unified, but all that I've heard follow the original plot closely. I favor the Geilgud/Richardson presentation from the 1954-55 BBC season and appreciate some of the freshness in the Merrison/Williams latter-day BBC production.
Several other Christmas-related Holmes radio dramas are in the archives. In "The Night Before Christmas" (Rathbone/Bruce, airing 24 December 1945), Watson dresses up as Santa Claus to entertain at a children's party; he finds himself entangled in a great jewel heist engineered by arch-villain Moriarty and carried out by "Lew the Lisper." Another is "The Adventure of the Christmas Bride," starring John Stanley & Alfred Shirley (airing 21 December 1947). Although not a Christmas story, "The Singular Affair of the White Cockerel" aired 28 December 1946, and the introductory banter between announcer Joseph Bell and Dr. Watson (Nigel Bruce) alluded to the recent celebration; that was the season Tom Conway portrayed Holmes.
For New Year's Eve, "The Iron Box" is an annual event for me. Actually, I listen to is several times throughout the year, because it's one of my very favorites from the long-running Rathbone & Bruce radio series; this one aired 31 December 1945. An alternate is "New Year's Eve Off the Scilly Isles" (Stanley & Shirley, 28 December 1947). Why not enjoy them both? They're only a half hour in length each.
Note that of all these year-end shows, "The Blue Carbuncle" is the only authentic Arthur Conan Doyle plot.