I can see that, and of course the man was devoutly Protestant. I expect that things beteeen him and Scotland might go sour slightly, and that he'll need to either banish Clifford or ensure that she and her family are bought off
Margaret and her sons would have to be bought off initially, but if they cause trouble then I have little doubt that they will be 'dealt with' in a more permanent fashion.
However the rise of Henry IX to the throne will have ripples on the international front, more than just causing a rift with Scotland. This will also end the hopes of Philip II of Spain to put a Catholic on the throne of England, after all the only Catholic Claimant left is Mary of Scots who is currently in English custody and will stay there until she dies, or does something stupid that forces the English to 'hasten' her end.
Now this doesn't automatically mean a Spanish invasion of England right away, Philip has to deal with the ongoing Dutch Revolt and doesn't need to antagonize England since that would guarantee that they would send assistance to the Dutch.
And Philip is involved in the latest episode of the Ottoman-Habsburg Wars, in the early 1570's the Habsburgs had managed to wipe out the Ottoman's ENTIRE fleet and seized Tunis, it wasn't until 1574 that the Turks had rebuilt their fleet and were able to take it back. So right now there's an occupied Tunis and looming Ottoman RE-invasion to deal with.
Then there's Portugal, granted Philip is the most likely heir of Sebastien I, Sebastien's death can be easily butterflied and Portugal has always had good relations with England (regardless of religious differences) as a means to keep Spain from just flat out invading and seizing Portugal for themselves.
So right now Philip II is in check, but it's a tenuous check.
France on the other hand is under the rule of Charles IX and would have seen the St. Bartholomew's Day massacre happen and the latest episode of the French Wars of Religion play out, and Charles IX's health is already in decline due to the stresses of his reign (Henry II of France has a LOT to answer for). I'm not sure if his successor, Henry III of France would have been elected to the Polish Throne in this TL, in any case he wouldn't keep Poland for long.
Despite all of this, an Anglo-Franco Alliance would be ideal for England... But I can't see it happening right away, and if Henry III of France dies without issue (which is VERY likely due to him being Gay) then the House of Valois ends and the French throne is up for grabs.
Scotland as I've noted before, is ruled by the Regent of James VI and they will be very unhappy as seeing their 'wee-King' being passed over for the English throne in favor of a minor lord. However this unhappiness will have to take a backseat to reality, and reality is that without good relations with England, Scotland is pretty much fucked since France is still reeling from the Wars of Religion and taking a hard turn towards the 'Old Faith', whereas Scotland is Calvinist at this point. England is the only major Protestant(ish) power that would see keeping Scotland in the Protestant camp as desirable.
The Habsburgs in the Holy Roman Empire are currently busy fighting the Ottoman Empire, plus there are all sorts of religious issues, and just general HRE nastiness brewing beneath the surface that could bubble up if the Austrian branch of the House of Habsburg decides to play games with England. They aren't going to do anything in the immediate rise of Henry IX, they have their own problems.
I could see the Catholic Church seeing this as their excuse to finally start throwing our Interdicts and Excommunications of the 'most vile heretics' that now rule England, but all that will do is just cause bad feelings and spiteful scoffing. The Pope's word isn't worth much across the Channel these days.