It's about a mile. You do know that all of the Downtown Diseny Resorts including the Double Tree provide free shuttle service to all 4 of the major parks, right? The DD buses are not meant to be park transportation for the DD resorts.
It's less than a 15 minute walk, but remember you need to go to a disney resort to catch a bus to DTD. If you are at epcot or mk it's easy to get to a resort then on to dtd. If you don't want to walk to the doubletree, just go across the street to the hilton and catch your resort bus as it's making the rounds.
Disney transportation is Disney transportation. It doesn't matter if you want to ride the busses, boats, or monorails. Jump on and have a great time.