Kind of what Susano said. I don't think Henry V could have possibly have tried to unite the kingdoms outright into one super-state. You had two very different systems of local government, law, and custom, and a lack of precedent besides; Normandy and England were never actually united. Add to that the fact that the succession dispute was already brewing during the time of Henry V, and granting the kingdom to a cadet branch seems a very judicious, very medieval way of handling things.
I never said one super-state. It would definitely be a Personal Union, like England and Scotland down the line. And they would be administered differently, for sure. Down the line, they would become more similar as parts of the government of one were adapted for the other, but they'd be different. For a start, I personally doubt that France would ever properly get over the powerful nobles - down the line, say in about 1700 or 1800, where England would probably be like as in OTL with important nobles with ultimately no local power, you'd likely still have a locally important Duke of Albret, Duke of Orleans, etc. Sure, they'd have their power to control their courts quashed, probably most or all of their law rights removed but they'd surely still control land ownership, have monopolies on certain trades, maybe even control things like their local mints, etc, totally unlike England.
And the considered English inheritance - Normandy, Gascony, etc, wouldn't be properly English either, even if Normandy did resemble England more than some parts of France. In fact, Gascony would probably consistently be one of the last parts in France to adopt every government reform and improvement, and certainly the last to accept centralising measures. Indeed, I think this idea of some parts of France accepting reforms at completely different times to others would be another defining feature of France over England. France would always speak French, England always English, etc. But the union of the crowns could last through all of this.
By the way, personally for my part I believe that if Henry had lived and continued to rule France, he wouldn't have ended up with what we today consider to be France. England was running out of patience with sending Henry taxes to pay for his armies, so too were Charles the Dauphin's southern French supporters tiring of the war and demanding that he enter into negotiations. More likely Henry would spend the rest of the 1420s consolidating his rule while
le Dauphin attempted to rebuild what he still had; any ensuing conflict would more than likely be ended by the Burgundians ending their involvement in the conflict and pushing for independence. Likely the HYW would end in a nominal English victory, with Henry's rule of northern France uncontested as he wins over the last doubters in France (there were actually a number of French nobles receptive to Henry being King) while the Dauphin's ironic title 'King of Bourges' would likely become more or less adopted as Charles' options and resources likely wouldn't be too favourable for a reconquest. So to my mind, any union of the crowns under Henry would see him take ownership of no more than everything north of Burgundy plus Gascony, anyway. In my mind I'm writing a TL around it, except that the later stages of it involves too much random chance for me to get away with posting
it on this forum.