”Am I right in thinking that QE2 was only #2 of England as QE1 was not Queen of Scotland?”

Going by this Wikipedia article, the use of Elizabeth II did cause some controversy in Scotland.

The issue arose again with the accession of Queen Elizabeth II, as Scotland had never before had a regnant Queen Elizabeth, the previous queen of that name having been queen of England only. Objections were raised, and sustained, to the use of the royal cypher EIIR anywhere in Scotland, resulting in several violent incidents, including the destruction of one of the first new EIIR pillar boxes in Scotland, at Leith in late 1952. Since that time, the cipher used in Scotland on all government and Crown property and street furniture has carried no lettering, but simply the Crown of Scotland from the Honours of Scotland. A court case, MacCormick v Lord Advocate, contesting the style "Elizabeth II" within Scotland, was decided in 1953 on the grounds that the numbering of monarchs was part of the royal prerogative, and that the plaintiffs had no title to sue the Crown.

It however seems that the generally accepted rule is that the highest number is used where there there would otherwise be differences between English and Scottish regnal numbers.