From reading the lyrics, it looks like it's to a similar tune as "Down the Road," the bluegrass song.

On the other hand, I have narrowed down some songs it could be. There's one widely-known one called Banks of the Brandywine (Laws H28). It's on the DT here or at this link: http://www.musicanet.org/robokopp/shanty/brandywn.htm.

Another song I have found is called "Oh Brandywine," which has several similar elements but doesn't exactly match the lyrics given: https://books.google.com/books?id=9L8eBgAAQBAJ&pg=PA39&lpg=PA39&dq=Brandywine++john+jacob+niles&source=bl&ots=4CUPzP729w&sig=46Q2ZguAXce8hRH5xCCh7AzIF5Q&hl=en&sa=X&ved=0ahUKEwik057HkaPLAhXL5CYKHeT_BREQ6AEIHDAA#v=onepage&q=Brandywine%20%20john%20jacob%20niles&f=false.

Another one, called "Blacksmith of Brandywine" is not that similar: http://www.thelyricarchive.com/song/169899-21326/Blacksmith-of-Brandywine

The lyrics also sort of fit to the instrumental fiddle tune "Brandywine" https://www.youtube.com/watch?v=lvlS0qZwuto

There's also a drum corps song called Brandywine Quickstep, dut I'll take a wild guess and say that's not what you're looking for.

There is a song called "Brandywine Valley" by Bob Paisley and the Southern Grass which I wasn't able to find online, so that could be it.