I generally like this idea, although I struggle to see how European politics would go afterwards with one dominating English-French state. One thing I like to constantly point out, though, is that Henry V's last words were regret that he didn't live to rebuild the walls of Jerusalem. A triumphant Henry going on Crusade is a TL I've thought about making a few times, with mainly shyness (and a perhaps unhealthy tendency to give pro-English bias) stopping me from doing so.
A lot of people posit that in the event of an Anglo-French Union, London would eventually become dominated by Paris and England would over time become little more than an autonomous French province - I'm not so sure.
There's no doubt that Paris was the better city in this era and France the larger and more affluent country, but I think people misconstrue the mindset of the English Kings by this point. For a long time, the Kings of England were very Frank-icised, for sure, but by the time of Henry V that had begun to be irrevocably changed. The English royalty and nobility had become used to being frozen out of France and had developed their own identity, borrowed from that of their subjects. The language of court was no longer French, for instance, and I think it's only a few decades before English becomes the language of Government too (from Latin). Also I'm pretty sure I read somewhere that the Treaty of Troyes included a promise not to annex French lands to the English Crown - considering the way that the English (royalty's) identity had been going, I can almost see this being subverted. I may be guessing a little here, but I can picture a French crown being stripped down, with the "traditionally-English" areas such as Normandy and Gascony becoming almost like Ireland was to England - autonomous areas with claims of independence, but ruled by a Lord Lieutenant in lieu of the King; geographically in France but not claiming to be under the French King - a rebuilding of the Angevin Empire in government but not in name. The Kingdom of France I can see thus still being very big and powerful, but being thought of as the King's second home, not his first. With a tendency for the important people at court to follow the King around, and with the tendency for the biggest institutions to go where they will be most noticed, I can picture a tendency for, say, some of the Parisian scholars to decamp (after some decades) to London, along with some of the more important merchants and such, and the important nobles to feel it necessary to go to London more often than Paris to be heard, even if they protest it. I'm not saying Paris would diminish to nothing, because it wouldn't, but I can see London coming on equal terms with it within a century. On a sidenote, I have to say I can see the Burgundians demanding independence in exchange for the part they played in taking France, and maybe even getting it.
I don't know, maybe I'm rambling, but I think an Anglo-French Union could work. I don't know what effects it would have on European politics, though. Perhaps the most interesting external affair coming up is that in something like 1450 the Countess of Provence dies with no heir, and in OTL leaves her (remaining) domains to the French King. This gives them a claim on Naples. An Anglo-French state with land in Italy would be cool, no?
