So... A bit more research (as I have also heard the buses start at 6 AM.
From:
https://disneyworld.disney.go.com/guest-services/bus-transportation/
Getting to Disney Theme and Water Parks from a Disney Resort Hotel
- Bus service from Disney Resort hotels begins 45 minutes prior to park opening and ends 1 hour prior to park closing.
- Return service to Disney Resort hotels ends 2 hours after park closing.
Getting Between Different Disney Theme and Water Parks
Please ask a Cast Member to assist you in planning your trip from one park to another. The most direct route may involve a combination of bus, monorail or boat.
Getting to Disney Springs
- Bus service from Disney Resort hotels begins 45 minutes before opening until 1:00 AM.
- Return service to Disney Resort hotels ends at 2:00 AM.
- One-way bus service from the 4 Disney theme parks operates daily from 4:00 PM until 11:00 PM or 2 hours after theme parks close, whichever is earlier.
So.... the answer is that it really "Depends". Apparently bus service starts 45 Minutes prior to park opening. Doesn't specify WHICH park and as such one must assume it is based on the "EARLIEST" park opening. This is from the Resort (or in our case the Fort) to the PARK.
However at the Fort, there is an Internal Bus system we need to contend with. I am pretty sure that it starts running before the other buses do as well. Your best bet, of course, is to check with a CM at the Fort (perhaps at Checkin) to ensure you have the timing down. As a rule, I figure on an HOUR to get to/from the parks and have been relatively safe with that estimate. Doesn't
usually take that long (once it took longer, but that was a unique situation) but it's a safe estimate.