As would I. I'd expect - somewhat ironically - for "French France" to centralise relatively quickly as the Dauphins (I'm getting tired of typing French Kings and getting confused with the King of France, so let's call them Dauphins for simplicity) attempt to react to losing their power base. I also anticipated the Dauphins annexing Savoy partly as the foreign policy equivalent of "I'm depressed, I'm going to get drunk and get laid" and partly as a means of expanding their resources to compensate, and such a large area of land without the typical French powerful nobles is going to help the Dauphins bash the nobles into line. In the north, however, with ultimate power ending up in London, I can see the French nobles who remain growing more powerful than ever - though I can't see English France disintegrating as some postulate that OTL France could have. Brittany would probably revolve in and out of the English orbit but essentially be an independent state under English influence - their loyalty would fluctuate and they might try and rebel but ultimately I think the English would keep them in line. I reckon they'd eventually just settle down to enjoying the freedom that the English granted them whereas the French were constantly trying to control them. But really that's another story. Champagne would probably be divided up, half royal demese (as IRL) and half for the defecting nobles. These nobles, plus the counts of Maine, Anjou, the Vendee nobles maybe (haven't decided who would get the Vendee), etc, would probably then keep a show of loyalty to allow themselves to become quasi-independent. But controlling much smaller areas of land, mostly with no coastline, they're going to be economically independent upon English France, so they will stay loyal - they only stand to lose power by defecting back to the Dauphin anyway, though that doesn't mean they won't try to intrigue with the south anyway. Normandy would probably be influenced by English, though I certainly don't think it would actually start speaking English. Gascony would be the same, but their language started off differently and they had a very autonomist regional identity anyway - Gascony might become a mini-France, with a whole load of sub-regional variations just as English France has a ton of regional variations. Each of these areas of France would probably develop a slightly different culture and linguistic variations, and I can't see there being any enforcement of a unified language - it just doesn't appear to have a benefit here. This acceptance of autonomous linguistic and cultural development would then probably spill over into the handling of Scotland and Ireland to a degree, and to the colonies later on too.
Burgundy would surely become an independent country. I can see Henry's brother (Humphrey, Duke of Gloucester) being more of an annoyance to an independent Burgundy, and I can see Henry stepping in to prevent the Burgundians slapping the Hollandais around (his main reason for arguing against Humphrey's marriage to Jacqueline of Hainaut was over the Burgundians' claim to Holland and Hainaut) so there's the potential for a very cut-down Netherlands in the English orbit, adding some Dutch flair into the language mix. Also, Gloucester was a bit of a power-hungry guy, and large parts of Frisia were uncontrolled at this point, so imagine him going off and conquering all the lands which - Papal guaranteed and everything - the Counts of Holland were entitled to. You'd have a County of Holland controlling the north coast of Germany virtually to the Jutland peninsula, and there's some real interesting language variations in there. It's great, there's a load of possibility.
Really, this is the one thing I'd really ever be interested in doing a TL about. However, my TL would be so biased and reliant on amazing flukes bordering on ASB behaviour that I don't think it would really ever be viable for anyone other than me.