What if the french cavalry had made a successful offensive attack against english troops during the Battle of Agincourt, winning the battle for the french kingdom and capturing the english king, Henry V?
It depends if the Duke of Burgundy is present or not. Due to the ongoing civil war between Armagnacs and Bourguignons, and while he raised troops for the battle, he wasn't that welcomed (his troops were) and one of the most powerful lords of France simply didn't participated to the battle. Possibly, his skills and leadership could have turned the battle. Of course, it would have meant that Armagnacs would have been ready to compromise their position and give their opponent a particularily good way to reassert its strength so you'd need a PoD that would at least divide Armagnacs on this question.
Without the presence of the duke, and Azincourt still being a French victory, it would be a major prestige boon for Armagnacs who would have definitely proven that they, not Bourguignons, were the party that could take down English forces (at this point, both factions insulted each other saying that they were in payroll of Lancaster). Not that both were that anti-English : Armagnacs did ceded several territories in south-west part of the kingdom to prevent Henry to ally with Jean sans Peur.
But at this point, you could see Charles d'Orléans and Charles d'Albret gloating endlessly about their victory and the capture of Henry V. I'm not sure it would be enough to see Paris definitely switching over Armagnac side, at least for a time; but it would certainly make Bourguignons looking bad. And giving many vassals of Jean did participated to the battle IOTL, it would be as well a personal setback for the duke.
It should be noted that, contrary to what you say there, Azincourt didn't began with a french offensive attack, but rather a french "offensive reaction as Henry placed his archery and fired the knights that had either to withdraw, or to attack in unfavourable conditions. I know it doesn't fit that well with the "French knight were stoopids munchkings laulz" but here we are.
just the French nobles weren't overly keen on listening to someone of low birth (who had drawn up said plan)
I'm not sure what you're talking about there. I mean, I know it's the Battle of Azincourt, but it doesn't seems to reflect any reality.
The revised plan drawn mere days before the battle (which actually took in consideration the real state of French forces) was probably if not made by Albret and Orléans, at least in their entourage. The tactics were, according to Barker, toughtful while the strategy is considered to be inconceivably bad, and French needing not to stick to the older plan but to actually revise it more. Which original plan was certainly not made by a low birth person and was nowhere as original you make it (it's the good old tripartite plan that was issued to more or less every battle).
If I had to guess, I'd say you may have felt victim to the old aformentioned narrative, complete with social stance "English army was about free men, and France was about fierce nobles". I'm afraid it should be nuanced a bit. And by a bit I mean a lot. And by a lot, can I advise you the excellent Agincourt of Juliet Barker?
Anyway.
If the battle is won by Armagnac-Bourguignon, you might see an uneasy truce with Henry V playing the role of prestigious royal hostage for a time, then a prize to be disputed when the conflict becomes hot again, whichever have Henry V under his control eventually signs a treaty with him against promises of support against the other (and probably some fortress turning to Lancaster trust, especially in Northern France).
If the battle is won by Armagnacs, then they would have little to gain delaying the liberation of Henry, probably exchanging his freedom against an ransom, return of most of North French fortress. Giving Henry V's skills, he would only turn to ask help from Bourguignons if Armagnacs are really dicks about the conditions of liberation, and these might know it. We could see, in return of a possible Lancaster help, some Lancaster or clients hostage in Armagnac trust. That said, Armagnacs could enforce harsher terms out of Henry, thanks to the clear prestige superiority they would enjoy for a while in France.
Would the english be expelled from France earlier than OTL?
Probably not, and it might even take longer.
Even in the case of a victory, we'd still pretty much back to Armagnacs vs. Bourguignons and while Lancasters wouldn't have as much resources to drain from Parliments and taxes to fund new campaigns, French would be a bit too busy to really deal with English pockets in Northern France and even less with a re-grew Guyenne in the South.
Giving the lack of trauma the Anglo-Burgundian victories in the 1410's and 1420's were for French (including Bourguignons), you might see a lesser drive to "bouter les Anglois fors de France", even if it would probably happen but slower ITTL.
That's not even mentioning that each side could promise a bit more territory to Lancaster, in exchange of military support.
Considering that Charles VI, The Mad, was still king of France at that time, how would the french handle an early victory in the Hundred Years' War?
Giving that HYW began in late 1330's, I'm not sure a victory in the 1410's could be considered as "early".
Anyway, the core words are
Armagnac-Bourguignon Civil War. As long it's not dealt with, a victory at Azincourt wouldn't be dealt with that efficiently. The better case IMO, would be an Armagnac victory without the duke of Burgundy. But it wouldn't resolve the war, as Jean sans Peur does appear with a man with better skill and larger political network in France proper.