Seneca check what I said. I was referring to both Williams and said "of both/the UK"; clearly the first was referring to William of Orange.
The current Queen was welcomed to the first opening of the Parliament as "Elizabeth Queen of Scots". The royal family use their Scottish crown titles north of the Border. Prince Charles uses the title Duke of Rothesay which is the title of the heir to the Scottish Crown. Elizabeth also went through a ceremony in St Giles with the Honours of Scotland, which are much older than the English Crown jewels.
In constitutional law, there is an argument that the Crown of Scotland can descend in a different manner to that of England, the legalities being based on Robert I with the Declaration of Arbroath, and on William of Orange and Mary II with the Claim of Right Act 1689 (The Declaration of the Estates of the Kingdom of Scotland containing the Claim of Right and the offer of the Croune to the King and Queen of England). The Act of Union united the Parliaments, not the Crowns. James IV and I was the first to speak of a United Crown of Great Britain and call himself that.
The current order of precedence in Scotland for women is The Queen; Camilla, Duchess of Rothesay; Sophie, Countess of Forfar; Anne, Princess Royal; Catherine, Countess of Strathearn; Meghan, Countess of Dumbarton; Autumn Phillips; Princess Beatrice of York; Princess Eugenie of York; Lady Louise Mountbatten-Windsor; Zara Tindall. All supposing that royal blood husbands are present. Funny that heirs' titles change in relation to a crown that supposedly doesn't exist. The royal arms also changes.