MK was pretty busy on the 1st. The key is to get there early as a lot of people like to sleep-in on the NY Day. The park was open until midnight and I thought a lot of people would clear our after the fireworks, but it wasn't the case. There was still a 45 minute standby wait on Peter Pan when we walked past it at 11 PM. But some things were walk-on (Splash and Haunted), and it was hit-n-miss (BTMRR 25 minutes, Buzz 50 minutes, Pooh 45 minutes, Space 30 minutes). We used Jan 2 as a down-day so I don't have any reports to give other than one person I overheard say MK was pretty busy. We were at Epcot all day today (3rd) and the crowds seemed a lot lighter today.
From my impression, the crowds on Jan 1 and 2 were probably like an average summer day crowd. They were managable if you know what you're doing, and don't want to blitz MK all in one day.