In this phrase, apparently the English queen's reaction to the Scottish vote to stay part of the UK was approval or satisfaction. Perhaps she literally said, "hmmm" in a happy tone, but it reminded the PM of a cat purring. "Down the line" means the telephone conversation.

David Cameron says the Queen 'purred down the line' over Scottish vote ... '.It means the Queen sounded like a happy cat when she was talking to him on the phone.