Are you sure it is completely closed? It is a 4 lane bridge with a center median barrier, we were just there for 3 days, it looks like they are able to keep traffic moving by only shutting down one set of lanes. However you may still want to avoid the area as they expect over 100,000 people to attend the celebration of the bridge opening in the 1970's. They have hundreds of vendors set up in the median and expect over 400 sky divers to do 800 BASE jumps on that day.
We did the walk under the bridge on the catwalk, WOW!
Or you could embrace the opportunity and check it out. The best campground we were able to find was 20 miles north at Mt. Nebo, Summerville Lake Retreat. Probably booked by now.
We tried the campground at Babcock State Park, but it was completely empty, with no full time attendant, and no cell service. I could almost hear the banjos playing. So we chickened out and ended up at another small campground at a kayak outfitter within a half mile of the bridge.
To answer your question, there will be detours posted well in advance of the area as there are no close by detours. The old Fayetteville road low bridge is in the shadow of the new bridge and has been completely reconstructed, but you wouldn't want do do that with anything but a car or SUV and it is one way southbound and has sharp switchbacks.
Bridge Day 2010 - BASE Jumping Registration and Event Information
Summersville Lake Retreat
I am posting this other info as others doing a search on the New River Gorge bridge may be interested.