We were at MK four times at opening (non-EMH days) (May 28, 31, June 2 and June 5) and the opening varied a bit each time.
Each time they let us through the turnstiles at about 8:30-8:35 am, then held us at the entrance to the underpass. On two days, they held the little ceremony/sang at about 8:45, then did a countdown. Once we got into the park, there was a rope at the various lands. (We were at tomorrowland each time.) Right at 9 am they did another countdown, and then let us into the rides. On Sunday when we were there, it went the same way, but a bit later. The ceremony was at about 8:50, so we got to the Tomorrowland rope at about 8:55.
On the morning it was raining, they let people all the way into the underpass, then at 9 am the characters came down to us to do the countdown, then the park was completely open. (We walked - didn't run - to Fantasyland and were the second family on Dumbo.)
One note: If you goal is to get to Fantasyland quickly, study your map carefully while in line. Unless it is raining you may not be able to walk through the castle because they may have the stage ready for Cinderallabration. If that happens, be sure to have your altenate route in mind. If you are a "Dumbo or Die" family, while you're waiting to get in, you may want to ask a CM if you'll be able to walk through the castle or not.