In 1420, five years after Agincourt, the Treaty of Troyes was signed between France and England. Henry V would marry Catherine, daughter of Charles VI, and become his heir. The Dauphin was disinherited.
Unfortunately, Henry V died in 1422, two months before Charles VI. Henry VI was proclaimed King of England and France, but he was an infant.
Bu what if Henry V and Catherine never had a child?
John of Bedford would succeed his brother as King of England. Could he have married Catherine, thus renewing the provisions of the Treaty of Troyes (after all, OTL's Henry VIII did marry his brother's widow)?
Could John of Bedford have become King of England and France? Could there have been a viable United Kingdom of England and France?