Looking on FreeBMD, there aren't many TOWERSON births in Berwick district, and cross-checking with census, they're all one family - here they are in 1861:
RG9/3887/58/10 Main St, Tweedmouth, Berwick Robert TOWERSON - Head - 38 - Grocer & Butcher - Egremont, Cumberland Catherine TOWERSON - Wife - 34 - Berwick Margaret TOWERSON - Dau - 9 - Scholar - Tweedmouth Jane TOWERSON - Dau - 7 - Scholar - Tweedmouth John TOWERSON - Son - 5 - Scholar - Tweedmouth Henry TOWERSON - Son - 3 - Scholar - Tweedmouth Mary TOWERSON - Dau - 1 - Tweedmouth Edmund TOWERSON - Brother - 41 - Lab - Egremont, Cumberland
A Robert TOWERSON married in Berwick in Jun Q 1851, and one of the possible spouses is Catherine BROWN. William Brown TOWERSON was born in Berwick in 1856, and died in 1857. (Catherine TOWERSON died in Berwick in Sep Q 1864.)
Although only the birth cert/baptismal record will confirm it for certain, I would think it very likely that Dora belonged to this family.