We just went in December with our 2yo son. My son has never been early to bed, early to rise (DW is not a morning person), but our lives were particularly out of whack in the weeks before our trip (in the middle of a move, big ice storm, etc.), and his bedtime had been creeping further and further back, so that he was going to bed between 12-12:30 and not getting up until 10 or so.
I had told my wife not to worry, he'd get worn out at Disney and get back on track quickly.
We arrived (driving) at WDW around 9 a.m. on a Saturday--son asleep in the backseat. We didn't even go to a park that day--we had breakfast at Boma in AKL, wandered around AKL looking at the animals, checked in at CR, rode the monorail, had dinner at 1900 Park Fare. Sure enough, DS was asleep by 10 p.m., and we had no trouble getting him up Sunday morning for EE at MK.
I agree with the above posters that EE was by far most important at MK, and useful even in slow season. Zipping through Fantasyland really gave us a head start on the rest of the day.