My suggestion is the tourist route: London-Cambridge-York-Edinburgh.

Return route: Edinburgh-Manchester-London

I don't advise you to travel to so many cities in just seven days. You waste all your time on traffic. If you can't see much, you have to catch the train. In fact, if you travel deeply, you will spend more than 3 days in one place in London. Edinburgh should also take more than 2 days. One day is enough for several other places ... especially Cambridge, which is a university town, and there is nothing to see after visiting the university.

Personally, I suggest London for at least 2-3 days. There are many places worth visiting, and most museums are free. Many streets are worth visiting, such as Regent Street, Oxford Street, Covent Garden, Hyde Park Corner, Notting Hill Gate and so on. Small streets are more interesting than these famous big attractions, but you need to do enough research before you leave, otherwise you won't find a place ... If you have a tight schedule, you can give Edinburgh a whole day. Be sure to visit the old city and see the old streets, buildings and castles. I didn't come for nothing (I don't think I have time to go to the scenic spots outside the city, and it's not worth going in winter and spring). If it is summer, it is strongly recommended to go to the Lake District (Loch Ness), which is very close to Edinburgh. It is said that York is very good, but I have never been there myself. I don't know what's good about this ... you can check it again. Manchester is an industrial city ... it's hard to say how much tourism value it has.

Big Bear