I agree do not rent a car, unless you have a specific need for it. In London, the streets will be confusing, they drive on the left (more confusing) and you’ll have a hell of a time parking it. The metro goes everywhere a tourist would want to go. Use that.
If you don’t use a tour company, look at sites like Viator, and plan some day trips from cities you’ll be at.
10 days in the UK you can cover a lot. I suggest splitting half the time around London, and the other half in Scotland, staying in Edinburgh. You can either travel by train between the two cities, or take a plane (about an hour flight). There’s much to do in both areas, but I like London better. Also, you might consider half your trip in London, and the other half in Paris, taking the Eurostar (chunnel train) between the two cities. Arrive from the U.S. in one city and leave back to the U.S. from the other. Between London and Paris, I still like London better.
That’s my two cents. I’ve been overseas 21 times since 2008, and over half of that in Europe. Not once have a considered a car.
Don’t get wound up with trying to save a bit on exchange rates. It’s not worth it. It won’t make any difference in your life. Just enjoy your trip.