It really depends heavily on traffic, but my guess is about 20 minutes or so.
Exit out of the Contemporary onto World Drive. When you reach the traffic light at Seven Seas, continue straight ahead. You will reach what we call the Jughandle. Please use caution at this intersection, and make sure to stop at the stop sign. The cross traffic will not stop.
Once you continue through the Jughandle, you will now be on Vista Boulevard. You will pass Fort Wilderness, and a golf course. When you reach the traffic light, take a Right onto Bonnet Creek Pkwy. You'll pass Port Orleans Riverside, and then French Quarter.
At the first traffic light, take a Left onto Community Drive. You'll pass Old Key West, a golf course, and the Disney Institute. When the road ends, take a RIGHT. You will now be on Buena Vista Drive. (The main road that DTD is on)
When you get to the major intersection, take a LEFT... You will now be on Hotel Plaza Blvd, where the Courtyard Marriott is located.