Yes, regarding the pillar box incidents in Scotland in 1953 - the article also mentions that happening. But, another question this all leads me to as far as the regnal numbering in Scotland is to ask which royal houses of Scotland were and are generally included in that numbering? Is it just the house of Stuart, or also the house of Bruce? Or others also that exised as well? So for instance, in modern times and into the future, if a member of the present British house acceded to the throne and his/her reigning name of choice was the same as a long-ago Scottish monarch, would that long-ago monarch be included in how the current monarch is styled in Scotland, or even more widely in the UK? For instance, when Edward VIII acceded to the throne in 1936, if he had chosen to be known instead by his actual first name of David, would he then have been King David II or higher in Scotland since there was at least one David on the throne there in early times? And then into the future with the present William of Wales, and when he accedes, will the pillar boxes and wherever cyphers are officially shown only indicate "WR" and not "WVR" in order to avoid another controversy?

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the article points out, such an event would mean the present Prince of Wales would become a future William III within Scotland. Is that correct?
No. There has only been on William, King of Scots. Commonly know as William the Lion. He reigned as King of Scots from 1165 to 1214. His 48-year-long reign was the second longest in Scottish history, and the longest for a Scottish monarch before the Union of the Crowns in 1603. There has been no other William who was King of Scots.

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the late Queen on her accession chose the higher numeral of monarchical name irregardless of what had come before in Scotland.
There was a huge outcry in the 1950s here in Scotland when it was announced the late Queen would be know as Queen Elizabeth II. There had never been an Elizabeth as Queen of Scots. When the original cypher EIIR appeared on letter boxes, some were blown up. Eventually the authorities caved and to this day you will only see ER on letter boxes. Police uniforms in Scotland also display ER.
As for what would happen if there was independence in Scotland I seriously doubt it would 'erase' the historic monarchical links of over 300 years. The numeriacy would be hammered out.