By January 2 I would stop wishing it. Only exception I make is when I visit forums weekly & I may have missed the Happy New Year thread so I do a Belated Happy New Year & add on something like, Hope this year goes well for people.
I tend to wish anyone I know a happy new year for around the first week or so and to those I have not seen since the prior year when I see them, I will wish them a happy New Year then.
After the first week of January, it doesn't make any sense to still hear it. So, it's when I no longer say happy new year to anyone. They already know it's the new year.