I figure you can just wish anybody the best from wherever you are coming from. It's the only solid ground you have. So whatever you celebrate, wish that happiness on others. If you don't celebrate anything, then I believe Mr. Dickens has created "Bah, Humbug!" and it is available for use in any denomination.
But I do like to note that the January 1 "New Year" is only one of several options.
I am particularly fond of the Chinese New Year, as it involves animals and great yearly cards and ornaments, not to mention horoscopes. So just to alert you, Chinese New Year is February 8th. It is the Year of the Monkey. Apparently the Chinese Zodiac timing starts a bit before the New Year, on February 4th, and there is a further refinement of 5 elements (one more than other Zodiac of 4: Earth, Air, Fire, Water). Here the five elements are Wood, Fire, Earth, Metal, Water. Apparently Air is not an essential element. Perhaps a mistake.
So Happy Red Fire Monkey Year!