I doubt it, since the “war with Russia” story arose in connection with the Crimean war, in the 19th century. Even though the story may be false, it derives its plausibility from Berwick having an ambiguous status.
Anyway, even though the Act of Union may have untied England and Scotland politically, they remained (and still remain) distinct countries, with, for instance, different legal and educational systems. It now seems to be established that Berwick is in England. According to its Wikipedia entry, however, although it had long been treated, in most circumstances, as a de facto part of England, that status was not formalized (and then only in an offhanded way) until 1972.