What day do you wish a happy new year?

Answer 1

New Year's Day.

The first time you see someone you already know after January 1st. For example - when you return to work after New Year Statutory Holidays it is normal to wish your colleagues 'Happy New Year". With your neighbours - it would be the first time you see them after the New Year. For family and close friends it is quite usual to phone, text or email them on January 1 specifically to say "Happy New Year" or if you had contacted them for a different reason the conversation would probably begin with "Happy New Year" anyway.

However, the further into January you are the less likely you are to say it - even if it's the first time you've seen the person in the New Year. It would be quite unusual to hear it said after mid-January.