Comparing 'Bramble Briar' with 'Constant Farmer's Son' there is little one can say conclusively. There are about 4-5 points of detail worth comparing other than the common plot and it's anybody's guess whether the writer of CFS took the inspiration from the original, from a later broadside or from an oral version of BB. All of the more detailed similarities are found in English oral versions. It is even possible, likely in a few cases, that CFS influenced some oral versions of BB, one noted above.

A summary of the similarities:
CFS Mary is in love with a farmer's son. In oral versions of BB her father is a farmer (a tenuous link at best)

In CFS Mary dreams about the death of her lover which is the case in some oral versions (instead of the ghostly visit)

CFS, like BB, mentions salt tears, gore, and her kissing the corpse.

In BB she hangs around his body 3 days and 3 nights. In CFS it's one day and one night. and when she has to leave, in both it's because 'hunger came creeping'.

What we do know about CFS is that it was written by George Brown about whom we know nothing other than he was one of the very few in the early 19thc who was allowed to append his name, like John Morgan' to the ballads he supplied. We also know the designated tune was 'Young Edwin in the Lowlands' on broadsides. Not only is it set in London but must have been first printed there.