England's under no threat at all if all four sons die. I mean sure, everyone seeks to preserve their own bloodline so there was a panic for a while after Henry V died OTL, but it wasn't a panic that England had no successor, it was a panic that Henry IV's heritage was going to be wiped out by a lack of heirs. Beyond Henry IV's bloodline, the inheritance is secure and clear. Next up is the bloodline of Henry IV's eldest brother, currently led by the 14-year Henry, Earl of Somerset, and he has three younger brothers (and two younger sisters). After them comes Cardinal Beaufort, another son of John of Gaunt (Henry IV's father) and then Henry IV's youngest brother, Thomas, Duke of Exeter, who was Captain of Honfleur during the battle. Neither he nor the Cardinal had children, but...they still could at this point. None of these men/boys were present at the battle.
That's a lot of heirs, and that's before you get onto the women. Now, the women are interesting, not because they themselves were interesting but because of Henry IV's accession to the throne. In order to remove Richard II's elected heir, the Earl of March, from the running, Henry IV in 1399 declared that all the way back to Henry III, no King of England was legitimately allowed to nominate or alter the succession to the throne. This technically instituted the 13th century semi-Salic inheritance, which prevented either women claiming the throne or men inheriting claims through the female line. This would make the future inheritance entirely open to debate, since that was a massive wrangle and somewhat hushed up in order for it not to create too much controversy (since about 4 or 5 Kings since Henry III, including H3 himself, had nominated a successor). It could well come down to interpretation of the law, or whether Henry IV was to be accepted as a legitimate King, if it got that far.
BUT none of this really matters as there's 4 male children of the Somerset line who would all have to die first for the female inheritance to even be contemplated here, and that's not likely to happen.
I don't see why that's so. The English hadn't taken Bordeaux yet, so it's definitely not under threat.