The only way a split could happen is if the POD was a lot further back and had one of the Kingdoms have a slightly different form of succession.
Like, let's say England was male-preferred, but a woman could inherit if there wasn't a man around who could (basically RL), but Scotland had (at some point) changed their laws to Absolute primogeniture, as in the eldest child (regardless of gender) inherits the Scottish Crown.
A union of England and Scotland could see a split happen if one of the Hannoverian monarchs has a daughter first, and then a son. Then again I don't see how such a manner of succession would (realistically) come about in either Kingdom in that era.
To my knowledge, until Mary, Queen of Scots succeeded (by lieu of there being no other royal males in the house of Stewart), Scotland preferred male-preference primogeniture, in that, even if the late king left a daughter, if he still had a surviving brother, the throne would pass to him. That said, England had a little bit of a less solid foundation, since while the house of Lancaster was descended in the undeniably male line from Edward III when they deposed Richard II, the duke of York claimed precedence when he rebelled, by pushing his descent from the more senior, but female line, descent from Lionel of Antwerp. Then the whole War of the Roses happened and finally Henry VII walked off with the crown after Bosworth. Now you can see the Tudors in one of two lights, but both involve female-line inheritance: if you take the Lancastrian line, Henry VII was only descended through a female line (and a bastard one at that), while Elizabeth of York was clearly the female line inheratrix of the house of York. Thus, no matter how you slice it, the Tudor claim to the throne was female line, which is only slightly removed from allowing a queen to succeed, I guess (Henry VII certainly feared it, since he insisted on Elizabeth being crowned
after he was king, to emphasize that he was king, not
king-consort).
So, for arguments sake, lets say Henry Frederick, Prince of Wales survives. He only has daughters by whoever he marries. Charles, duke of York and Albany, has sons - however, there would be a generational disparity most likely (since Charles was nearly a full decade younger than Henry). Henry becomes Henry IX of England and Ireland, I of Scotland, but when he dies, his daughter becomes Queen of England and Ireland (in theory), while his brother becomes king of Scots.