England in the 1400s
Henry VI, Part 3 is set toward the end of the English Wars of the Roses. The play ends with Henry dying and Edward securing his seat on the throne, with an heir to boot.
But even Edward's days on the throne are numbered: soon, his brother Richard will make a pass for it and set off a fresh set of battles. This play shows the end of the York vs. Lancaster argument that began in Henry VI, Part 1, because after this, York (Edward) will be fighting with York (Richard). These battles are still technically part of the Wars of the Roses, but they're confined to the York family.
From a historical standpoint, the events in this play were going down around 1471.
We jump all over the map (of England) in Henry VI, Part 3 and even make a quick trip to France. Most of the play takes place in London, York, or somewhere in between. London was the seat of royal power in England, so it makes sense that this play about royal power would be anchored there. York is the hometown of—you guessed it—York and his sons, so staging the action there is about who has the hometown advantage.