There are so many things to do in all of those places it is difficult to know where to start. First, I’d look at the sticky at the top of the page on free things to do. Second, I would investigate the heritage pass and several others that are offered to see which ones fit your desires. I’d also try to delineate what it is you want to see.
Third, I’d consider writing to get free tickets for the “Ceremony of the Keys” which needs to be done in advance. It is night time trip through the Tower of London. Fourth, I’d investigate and get tickets on line for stage shows playing in London when you are there. For this, consider using one of the concierge services with Visa or Mastercard if you have a signature card or even consult the concierge in the hotel when you get there but by then it may be difficult to get exactly what you want. Fifth, I’d look into tour companies that will take you outside the city to places like Stonehenge. Personally, I would first take one of the trains and then hookup with the tour companies rather than trying to get something right out of London. We took the train to Bath and then took the Rick Steves recommended tour which was excellent. I would get Rick Steves book on England and Scotland. Sixth, I’d get familiar with the various train stations in London and learn where trains from a specific station go to. There is so much more ie. Albert and Victoria museum is great. Churchill war bunkers (shown in the movie Dunkirk), several Harry Potter sights and tours. Don’t forget shopping at Herrads-even if you buy nothing it is a sight onto itself. What about the British Museum? It is awesome. There’s always the London Wheel. How about Stratford on Avon? So much to do, so little time.
Scotland is also terrific. I’d stay in Edinburg. There is a nice starwood hotel in sight of the castle. The Sterling area is the William Wallace Tower and so much more.
Are you going to the D-Day beaches in Normandy? In Paris, again get familiar with the train stations and their destinations. Obviously, I wouldn’t rent a car- you don’t need it. Of course you will get advise from others who couldn’t read a timetable if they tried that you do. In London proper, a car is totally worthless. If you sent most Americans to the moon, I think the first words out of their mouth would be “Where can I rent a car”. Between the trains and the tours it is easy to access everything. Paris is another 5 paragraphs. This should give you a start.