Possibly. I agree that your premise is possible, though not so much for the reasoning. Well, that quite possibly could be the reasoning too, but by this stage, it wouldn't make sense for the Burgundians to up and turn on Henry unless they were willing to submit to the Dauphin in Bourges, because the other alternative would essentially be a three-way war with virtually equal-strength combatants, and that is a war they would be far from comfortable at their chances of winning. They historically did submit to the King of France, but they lost their shot at the throne that way, which is why I disagree with your reasoning. I think that they would have to wait for Henry to have decisively put down the Angevin faction before turning on Henry.
The reasons for Burgundy's submission to the King of France would not apply to TTL, unless Henry V were to be killed in a TTL battle. And John, Duke of Bedford, did not remarry.
Just liked it has happened in the Anglo-Scottish dual monarchy?
Don't think the Scots were speaking Pictish in the HYW, much less at the establishment of the Stuart Dynasty in England. A common language helps to cement two peoples. Especially when the two countries are only separated by Hadrian's Wall, not the English Channel.
But my actual point here is that I honestly am not convinced that Henry could complete his conquest of France in his lifetime. He was running out of resources, the Angevins were about to bounce back (let's face it, Joan of Arc would not become Henry's mistress but would entirely possibly become prominent in TTL - the POD is too close to when she started making sounds) and the English were starting to lose Generals.
Thank you for being one of the few not to see Joan as a mere mascot, and not to assume that Henry V would simply crush her under his heels. The fact was, Bedford himself never faced her in battle OTL. It was only Glasdale and Talbot who went up against her head to head.
It always seems to be assumed by ATL writers that a surviving Henry V would have a sudden flash of inspiration from the moment of first hearing of Joan's story, and bring the whole of the reserves of the English Army forward into battle and at DEFCON 1.
Personally, I can't see him as reacting any differently than Bedford. In short, seeing her as an amusing annoyance (a sign that Charles VII must truly be done for, to consider using a green peasant girl to lead his army), and a "law enforcement issue", by ordering her arrest (as OTL) before she could reach the Dauphin. NO ONE took seriously the idea that she could contribute martially on the battlefield. That is, until after the Fall of the Tourelle.
In addition, England was running short of willingness to keep going and while I believe that Henry could keep them willing to pay for the war, the subsidies would reduce while the amount of men in the field would need to increase (from garrison duties as well as serving the King's army in the field). In the meantime, Burgundy and Brittany were becoming more unwilling to continue the fight too, and Humphrey Duke of Gloucester (Henry's brother, think that's his name) was bound to start causing trouble when he married Jacqueline of Hainaut, the besieged ruler of several territories the Burgundians desired - something I think he would still do, probably forcing Henry to take his side eventually. Far more likely that the end result is the war grinding to a halt with a few big victories for the English weakening the Dauphin's cause, to be put in check by a myriad of little successes when smaller French forces chipped away at the places Henry wasn't.
The evolution of French artillery was also about to change drastically the way war was fought. Castles and cities that had held out for months and years would fall within days of opening bombardments. Both sides of the HYW held the advantage at different times, but once heavy fortifications had become truly vulnerable for the first time since the introduction of the trebuchet, mobility was reintroduced into the war. That gave the advantage to the side with the better cavalry. The French. And once the French finally learned (for the second time, Charles the Wise learned it first) NOT to charge prepared English Army field fortifications, but to bypass them, they seized the initiative at last and never lost it.
These practical considerations of new military technology and tactics developed by the French (not unusual for the losing side to do this
) tell me that Henry V might well have found himself in the situation of Napoleon in the closing days of the French Empire. Beating the enemy in the field wherever HE was, but losing everywhere else. As his enemies intended. And as you yourself suggest for Henry.
The likelihood in my eyes is that France eventually gets split in two, probably more de facto than de jure. What the Burgundians do next is unknown but they would probably turn on the English eventually. By this time I suspect Henry would have developed more of a French backing, with some French nobles at his side who could only remember times of English rule, and the Gascons an ever-present ally, but when Henry dies I see English rule being pushed back in the centre of the country. From there, it's unforeseeable - the English could either cement their control over Paris and the north, or eventually suffer the same defeat they suffered in 1453, just much later on. It's possible, if unlikely, that after another century (or multiple) that the English could drive south again and finally win the war by capturing the entire country, though I can't see that happening in anything under 100 years of Henry's death.
Purely from a physical/geographical standpoint a "Burgundian" nation could never have survived. Squeezed between France and the Germanies, with its stomach in Flanders, its heart in Dijon, and its head in Paris(?), such an entity has no real center of gravity. Never mind military defensibility.
As to France? Unless Joan of Arc is captured, Charles VII is murdered, and Henry V outlives his own son (leaving his country open to a different kind of Wars of the Roses?), he won't be able to swallow up such an enormous mouthful of the whole of France in his lifetime.
Even if Henry V lives to a ripe old age, he'd be 67 in 1453. That's ancient by the standards of those times. And that is just about the time that the future Henry VI, described as a "lame" king, which he certainly was, became suddenly near-catatonic. This was a man who now couldn't even wipe himself, never mind rule a Dual-Monarchy. And there were powerful forces that wanted to keep him right where he was despite his newly acquired imbecility.
As to Henry V having more children? Neither of his brothers had surviving issue, and he himself had only the one boy Henry who grew up to become mindless at the age of 31. It looks like the Lancastrian line was thinning out.