Brought up in Scotland I know all the traditions for Hogmanay but apart from eating steak and kidney pie on Ne'erday I don't know of any connected to New Year's Day.
In Denmark too, some families hold that the house should be clean and tidy, the laundry done, dried, ironed and put away before midnight on New year's Eve.
A long time ago, there was a superstition that said that whatever you did on New Year's day would be repeated frequently in the course of the year.
As New Year's day has been a public holiday for the last couple of hundred years, no-one bothers about that superstition any more, but I would never dream of doing the washing on New Year's day, as there isn't any needing to be done.
The old superstition doesn't work, I know for a fact, because I always listen to the New Year's concert from Vienna, and much as I would love to, I have never heard Wiener Philhormoniker every day of any year!