Can't speak for anything past 1-2. And know that this comes from someone who abhors crowds, so it's biased (see my book, "Why they should close Disney for me when I visit").
That said, lemme tell ya. I've never, ever seen anything like NYE at Disney. We absolutely hated it. I should point out we were with friends who thought we were nuts (note, they also talked me into Times Square eight years ago). We managed to get into MGM around 10:30. I'd had it by noon. I think MK was closed by maybe 2 PM? And Epcot, which is my fallback (never have I not found room to breath in Epcot) was open, but packed. We left at 6 PM when the line to get a Beaver Tail was over fifty people deep. And that was the shortest line we saw.
We then made a really bad mistake (because the tickets were free) and went to PI. I mean, if you're not into crowds (and I know some people dig them), just don't go there. It's nasty. Only time I've ever been to PI and never made it to the AC.
To make matters worse, we were there to see the Nittany Lions in the bowl game, and we got our butts ground into sausage.
But after the game, we went to Epcot. On January 1, the crowds seemed very, very light. The next day we were in MGM and Epcot and it was barren. Made me wish we had time for another day.
So, if you can get into crowds, enjoy NYE. If not, stay at the hotel. But after that, you should be ok. A "first week of January" trip should yield you a good trip.
Pat