"I own a working Edison Amberola that looks pretty much like the one in this picture. The cylinders look like those in the picture. They're made of a hard blue plastic-looking material over an inner cylinder that looks like clay. These are not "wax cylinders", are they?"

Yes, they are

"The National Sound Archive appears not to give a collective flying fuck about folk music and should not be entrusted with anything like this."

Up to about 20 years ago this might have been the case, but since then, when Lucy Duran joined, there have been changes for the better. I for one am very grateful for somewhere to have deposited our own collection of field recordings - it means they are going to be around much longer that we are.
The International Music Department recently completed its 'Bright Golden Store' project which added a number of collections of British traditional music to its holdings - visit its website sometime www.bl.uk/collections/sound-archive