There is also the possibility that they go with dynasty names and in the Renaissance they could call it Tudor instead of England. It wouldn't be my preference.In that sense, perhaps England could be Renaissance and leave Britain for the Industrial era, which could be transcended to the Modern era with little issue. But the thing is England does have a significant presence in the Middle Ages, and has iconic stuff like longbowmen. Redcoats and Royal Navy-related things could be subject of either the Renaissance or Industrial eras, as British more than English.
Personally I would go with Anglo-Saxons then England, then Great Britain/United Kingdom for Medieval/Renaissance/Industrial.