a man named Pukunui should know Pukeko in a Punga Tree!
Mele Kalikimaka was once kinda popular (though Bing and the Andrew Sisters are a bit dated now )
Aotearoa New Zealand
Christmas in New Zealand - "the time is right for barbecues and swimming across the bay"
country singer Dennis Marsh put out a Christmas Album
Te Harinui - faith song about the first Christmas Sermon in NZ 1814 "not on a snowy night, by star or candlelight"
and Ronan Keating (and Julian Denison) did a Summer Wonderland song as an Air New Zealand ad
most other NZ Christmas songs are child focussed, or traditional carols, a few country singers do Christmas covers though. Last Christmas and Feliz Navida also get a lot of air play
Australia has a few Christmas novelty songs
Christmas Where the Gum Trees Grow "there is no frost there is no snow, Australia is hot, cold and frosty it is not!"
Aussie Jingle Bells - novelty version of Jingle Bells "its Christmas time and I am in my singlet shorts and thongs"
Kevin Bloody Wilson did a popular novelty song about Santa not bringing him a bike- but its not Grandma friendly