You land in London and the airport still says United Kingdom. The tourist boards do not. England, Scotland, Wales and Northern Ireland are separate ideas of distance, weather, and what a day trip means. A week that tries to collect all four is a week of trains and very little sitting still.
The United Kingdom page is the map. Tower Bridge is one photograph in one city. Edinburgh Castle is another country on the same ticket, and a different train, not a neighbourhood.
Pick one of the four and a second city inside it. The passport is shared. The week is not.
