In Spain, you say Happy New Year as Feliz Ano Nuevo. Here are some more ways to say Happy New Year in other countries:
Dutch - GELUKKIG NIEUWJAAR!
Eskimo - Kiortame pivdluaritlo
Esperanto - Felican Novan Jaron
Estonians - Head uut aastat!
Ethiopian - MELKAM ADDIS AMET YIHUNELIWO!
Finnish - Onnellista Uutta Vuotta
French - Bonne Annee
Learn Happy New Year in different languages here.