What is the history of an old double barrel 12 gauge shotgun?
If there are no other marks, it will be pretty difficult to identify the maker or exact age of your shotgun. I can say it was definitely made after the 1914 patent date, so it is considered a modern gun by collectors. I can also guarantee you that it was not used in any war. There haven't been any double barrel shotguns used by any army since the US Civil War, and only a few then. Unless it has fancy engraving, carving, or silver-plated parts, it's neither rare nor valuable, but as your grandfather's gun it should be priceless to you and your cousins.
I have know nothing about J.N. Scott, but there is another thread on this page that has some information.