We were there this past NYE with a then 21 month old and 3 year old. We had a FABULOUS day at the AK that day.

We knew that everyone else would want to be somewhere with midnight festivities and that the MK and Epcot would close at capacity early in the day so we avoided those completely (well, we did go to Epcot first thing but we did Journey into the Imagination, Living Seas and Spaceship Earth then headed to the AK so we weren't there long and NO ONE was in those areas b/c they were all at Mission Space and Test Track

). AK closed at like 5 or 6 so not many people headed there. We pretty much walked on to everything that day. When we decided to do the safari ride at like 2:00 we waited in our longest line which was about 20 minutes. We had ridden it earlier with a less than 5 minute wait.
When we left AK we had a dinner PS at Cape May Cafe at the Beach Club. I had scheduled our PS to be at about 10 minutes after the start of Illuminations at Epcot (the first showing, which was an early one. I can't remember the exact time but we left AK at around 5:30 and had just enough time to catch a bus to the Beach Club, clean the kids up a bit, meet up with my parents and steak out a spot on the beach behind the resort to watch Illuminations). We watched the upper fireworks from the beach at the Beach Club. A better view would have been to walk around to the Boardwalk but we were pooped at that point!

We had just enough time to was everyone's hands before we were seated. We took our time having a nice dinner and brought the kids back to go to bed. It was our only day we didn't head back for a nap mid day so they were tired too. They napped in the stroller for a bit but it's not the same. They got to bed relatively early which worked our perfectly b/c we were able to be up and out the door early in the morning New Years Day and get a BUNCH done at the parks while everyone else slept in after their late night!
DH and I had contemplated going to MGM after getting the kids to bed (my parents were with us so we had free babysitting) but we decided to crash early too so we could get that early start. We did MGM New Years morning and Epcot that evening. Doing that combo we saw the 3 big things the kids wanted to see at Epcot and all of AK with no waits on NYE, then did everything we wanted to see at MGM and the rest of Epcot with no waits on New Years Day. We couldn't have asked for a better plan!
We'll be back again this year and plan to do something similar again!