The last 3-years we've done NYE there, including the week after . (Doing it again this year, Dec 30 - Jan 10)
It seems they always start taking decorations down about the first weekday after new year's. So this year that would be Thursday Jan 2nd.
Most of the park decoration are removed within a week or so.
The Osborne lights come down last My guess this year is they'll probably start coming down anywhere between the 7th and the 9th, just looking at the calendar and seeing where the weekend days are. It takes them a good week to get them all down, but they'll be turned off as soon as they start.
All are just my estimates based on past experience.