I'm in no way suggesting that the other posters are wrong, but I'll tell you my experience from NYE 2001. DH & I had dinner at Yachtsman Steakhouse at approx. 6:30. We had a pretty extensive meal, so my guess is we left by 8:30 or so. We walked over to the International Gateway entrance to EPCOT and walked right in. Now, we only went over to UK. Yes, it was packed, but it was an incredibly JOVIAL crowd. Not nasty, mean, obnoxious, etc. We really enjoyed the atmosphere, watched the "Beatles" group and just hung out until approx. 10 pm or so.
We are returning this year to the Dolphin, God Willing!!!, and plan to do the same thing. I'm sure that twds Future World with kids would be no fun! But, my suggestion is the areas closer to the International Gateway area. I would think they would be a little less crowded. Also, the year we were there - 2001 - was right after 9-11. We thought it would be empty, but most places were filled to capacity. Many of the CM's commented that it was the most crowded Christmas Week that they ever remember! So, that's my .02.
Whatever you do, it will be magical. Just be flexible and ENJOY!