I will relate my own personal experiences with NYE at both MK and Epcot. I know that a lot of people will not agree with me. They go with the "get in early and stay all day until midnight" philosophy. I simply cannot do that. It's way too long of a day for me.
I get to the park of my choice as early as possible. Last year it was a bit more problematic since we did the evening EMH at the MK and were there until 3AM! While it was fun, I won't do that again! Anyway ... 7-8 AM is the best. You can ride a few rides and get as many fast passes as possible. Remember, you can almost always use a fast pass after the alloted hour. We sometimes make an ADR for late morning or early afternoon and then around 1-2PM we get out of the park. The MK is usually closed to new guests about this time and will remain closed until after the parade. I have heard that Epcot will also close in the mid afternoon, but the International Gate has always been open when we leave.
We spend a few hours back at the room. We nap a bit, nosh a bit and maybe swim. We return to the park between 5 and 7 PM (depending on our dinner ADR). We have never had a problem re-entering. Once we return we grab hats and horns, use some of our stale fast passes, eat dinner and enjoy some of the live entertainment. We stay for the fireworks and then head on home.