Agincourt took place on October 25, 1415. By October 28 or 29 (most likely the 29th) news of the victory had already reached London. It was common for victories like Crecy or Agincourt to be announced in the form of an official newsletter from the king, but although we have examples of such letters from other campaigns, there is no surviving official documentation of the official announcement about Agincourt. The first official celebration in London was a religious procession to Westminster Abbey led by city officials and joined by the citizenry. Because we don't have the official announcement, we don't know exactly what the news might have been There is some similarity between several different recordings of casualties, prisoners taken, and the overall course of the campaign in local records, which may show that there was an overall "official" account of the entire Agincourt campaign that Henry V produced and disseminated.
A random commoner would likely not be too aware of the particulars of these records, however. The official accounts, with their lists of important French nobles killed, seem to be aimed at a more elite audience. It is doubtful that the particularities of whether the French Master of the Crossbows survived Agincourt or not would matter much to a commoner. However, a commoner might have been present at a public reading of the victory announcement, or heard people talking about it after the procession. Someone in London would of course be the first to hear the news and dockworkers would be making preparations for the army's return. People with connections to the army- sons, brothers, fathers, husbands, etc. who had served- would of course hear from their returning men more details about what happened. But people who saw the force disembarking would at least be able to tell that the army was returning without too many losses. What rumors about the battle and the events of the campaign were flying around, we will never know. We can probably be certain there would be a lot of soldiers all trying to claim their share (and then some) of the glory, though! Such is the nature of soldiers throughout recorded history.
We have no evidence to indicate how quickly people in farther flung parts of England heard, but it would depend on how much communication there was with the rest of England in general. Major cities like York would probably get the news as quickly as someone could travel from London to York. Far-flung outposts on the Scottish border or isolated Welsh villages might not be aware for a month or so. The English-held city of Bordeaux only had the official announcement of the victory read on November 23. Unofficial communications had probably long since arrived at Bordeaux, but apparently the city government was waiting to make the official statement to the people until they could arrange to send envoys to London in order to congratulate Henry V in person.