He would be King before Elizabeth became King. Either way, James VI and I decision to settle Scots in Ulster would be butterflied away.
Settling a rebellious subject people on a frontier, thus making them loyal, is a very tried, and very true, approach to dealing with rebellious subjects. I don't think the Scots were particularly rebellious, but a loyal settler population was necessary to make sure that Ireland could be properly (from the English perspective) dominated by the English crown.
No Scots in the English Army would change the North American war with France. If you change the North American War, do you butterfly the American revolution? If so do you butterfly the French Revolution? If the Tudors decend into absolutism then do you have an English revolution? I just cant see a larger Empire than OTL British Empire though for any of the Great Powers.
I think that you're missing some rather large chunks of much more important short term stuff that will then affect things like the Anglo-French colonial rivalry.
The reason Henry VIII did not attempt this was due to the Scottish Alliance with France. If Henry or his heirs even attempt to invade Scotland then it means war with France. Such a state would only drive the Scots further into the arms of the French.
The French can't really affect English policy vis a vis Scotland, because they don't care about Scotland. The French cared about making sure that the English could be distracted from burning their country down from time to time with border raids. So then the Franco-Scottish alliance is based on a mutual dislike of the English.
So if Henry VIII's heirs figure out that war with France with the aim of taking the French crown is un-doable (and the 100 Years' War proves that it is), then they focus on things much closer to home. The French are occupied with wars against the Hapsburgs in the Low Countries and Italy. If the English aren't making trouble with ill-fated expeditions against France then they can successfully subjugate the Scots.
Naw. The Stuarts ruled Scotland for hundreds of years. The only reason Mary and James VI were heirs to the throne of England was due to the marriage of James IV to Margaret Tudor. The English claim of the Scottish throne was not as secure as the Scottish claim to the English.
But the English claim is being advanced by a well-run, trending absolutist regime, against a divided Scottish regency. Henry VIII's son (shall we call him Henry IX?) will be a Protestant with very close familial ties to the infant King of Scotland (I'm assuming that Mary, Queen of Scots pretty much proceeds the same way).
The kind of scenario I'm imaging is that Henry IX finds his familial relationship to the Scottish royal family a great aid in taking the country over. Following some thing along the line of OTL events Henry IX finds he needs to use military force to make sure that his cousin's interests are protected.
Or Mary, Queen of Scots', French husband doesn't die, and the Tudor regime just conquers Scotland. Scotland, without enough aid from France to fight off the English, is conquered. Though French Kings continue to style themselves "King of Scotland", the title is actually held by the English.
The lack of actual title to Scotland is ended as an issue when Henry X is crowned as the "King of Great Britain" upon his accession to the throne achieving his father's vision of a united, absolutist regime.