New Zealand has a bunch -- "Pūkeko in a ponga tree" on the secular side, and "Te Harinui" on the Christian side, come to mind immediately.

(I've also done a filk "I'm dreaming of a bright Christmas" but it takes time for these things to catch on to the whole country....)

I feel like we're accreting Christmas traditions that are appropriate to our shores and climes, but it is a pretty slow process. And of course the Coca Cola Santa still reigns supreme.