Ex historian so I have a vested interest.... I think that there is history that people need to know - ie the history that explains how we are the society that we are today and the history that is good to know - often that which explains the physical bits of the past that we see around us. The former would include (by no means comprehensive):
History of English relations with the rest of the UK - ie Edward I and the conquest of Wales, Union with Scotland, colonisation of Ireland
History of Parliament -Anglo Saxon law, introduction of the feudal system, Magna Carta, Simon de Montfort, Civil War, suffragettes etc etc
British Empire
Medieval guilds, Industrial Revolution, Labour movement etc
The good to know bits would include:
Roman Britain, road system, major towns etc, Hadrian and Antonine Walls, trade/colonate/patronage
Norman conquest, Doomsday Book and the building of castles to control first England and then Wales etc, medieval affinities and the Wars of the Roses
Conversion of Britain, Reformation, anti-Catholicism, Armada, Gunpowder Plot etc,
Redistribution of wealth and power under the Tudors, agricultural reform, enclosure, development of the English Country House,
Urbanisation