The "next day" rolls over at 6am.
If a room in your category is available, you can go right to it if you don't care what building/floor/area it's in. If it's what you booked, Disney will let you have it if it is available (for example if it was not used that prior night).
Note I said "if it's available".
If it's not available, you have to wait. Or pay the diff for an available room for the length of your stay. The room has to be turned over and cleaned.
I would stay someplace near Orlando overnight if I were you and roll into WDW the next morning after you have had some sleep. You might end up waiting still (although you could store your bags and head to a park if you wanted to in that case).
That 6am rollover time can work in your favor. If you can book that same level of room as a 1 night reservation (even if it is at rack rate) and you might get in at 2am, it could be worth it to book that extra night and roll in during the late night/morning hours and then sleep in the next morning. That's the other side of the coin. As long as it's the same level of room, you won't have to move when your original multi-day reservation comes up for check-in. Just tell the CM about it when you arrive.
Some people might have a discount of some type and may not be able to add the extra night on the same discount (without losing the discount on the original reservation). If having a room when you arrive early in the a.m. is important to you, book it as a 1-night at the same type/level at rack. The peace of mind and sleep is worth the price.
Bama Ed
PS - if I had to bet $100 of your money, I doubt a room would be available early at 6am for you at your hotel. That strategy sounds good in theory but the odds are WAY in favor of you having to wait to get to a room.