I visited the Bohemian Alps Campout in Verdigre at the end of June. The only places 12 was closed was just east of Niobrara. There was a detour sign to the west of Niobrara, but FWIW a local said a couple miles of easy gravel near the state park would take you around it. I didn't investigate that, but wish I had known, as I really enjoy that stretch.
You can take some back roads from east and south to get into Niobara -http://tinyurl.com/3gkraaf. Highway 14 from Verdigre to Niobrara is open.
There is a nice little paved county highway which runs west out of Verdigre for about 10 miles. When pavement ends (at Opportunity, NE), follow the main gravel road north 6 miles and it turns back into another paved road which goes east over 281. Kind of neat road -http://tinyurl.com/44agr8x