Mark,
If you have a sharp pair of diagonal cutters you can get in between the boards to cut the pins and then remove them one by one. I would suggest that instead of trying to remove them all together. You are correct there is not room for the Nano and a socket, at leas that I could find. I went ahead and mounted mine back where it came from on some header sockets but I intended to separate my display and Nano anyway and use a serial display. If you want to keep them together put the sockets on the other side and when you solder the pins into the new Nano have them stick? up instead of down. That way you can mount the Nano on the other side and it still have the correct pin out. I would have done that but all the Nanos I had at the time already had the pins soldered in the conventional way. Here is a picture of what mine looks like now.
Jeff
