The earliest copy in print (1740-1750) only says 'town'. (He sent his man down through the town to the place where she was dwelling).
In Scarlet Town is found in print in 1765.
All the others come later.
Quoting A. L. Lloyd in his "Folk Song in England," "'What county does that song come from?' is a question commonly put bu innocent amateurs. They inquire in vain. Sundry versions of 'Barbara Allen' give the young lady's dwelling place as Scarlet town, London town, Quelick town (wherever that may be), as Reading, Newbury, Newry, Dublin, ..... no doubt with a view to making the ballad interesting in whatever locality it is sung."

No one knows what town the singer was referring to when he sang 'Scarlet town'.