The object of the nutcracker, depicted as a soldier, comes from the central prop/character in the the 1892 libretto, The Nutcracker by Marius Petipa (itself adapted from an earlier work by E.T.A. Hoffmann, The Nutcracker and the Mouse King)

Petipa's story begins on Christmas Eve, hence the connection.

I can't say when the earlier E.T.A. Hoffman work takes place but I imagine it too is over Christmas.