Yeah, it should be fine. Nothing but snow and fires close the road. As long as you are looking to kill a day start at the Museum of the Cherokee Indian right there in Cherokee. Then make a stop at the Oconaluftee visitor center at the entrance of the park. A not to be missed stop is Clingmans Dome, the highest point in the park, great view. Lots of water falls to see along the way if you don't mind a little hike so bring the hiking boots. Then you have a stop at Newfound Gap, Then the down hill side of the trip into Gatlinburg so you can either make a stop in town or head west at the turn off at the Sugarlands center to go to Cades Cove and Townsend area. You can pick up I-40 by way of Rt. 321 and cruise on into Nashville.
Mileage wise its only 35 miles from Cherokee to Gatlinburg but the speed limit is 35 all the way thru the park and depending on traffic it will take and hour if you drive straight thru, throw in a few stops and it can be who knows how long. All up to you.