Your plan sounds good, given the amount of time you have. Of course, each traveler would have their own personal preferences and do it a little differently, perhaps.
You may feel a bit rushed trying to see the Cotswolds as a day trip--but just prioritize and see your favorite sights first and see how it goes.
5 days in London is not too much, IMO. Even with 5 days, you'll probably only see 1/2 the things you want to see--again, prioritize.
If you've never driven "on the wrong side", strongly consider renting an automatic transmission, the standard car rental there is a non-automatic standard shift. My personal experience picking up a rental car in Bath is that it's not sufficient to just check the box for automatic transmission on the on-line car rental form when you make your reservation, because it's likely there isn't one automatic transmission on the car rental lot when you get there. A couple of days before you depart, call the UK number for the car rental location, and say something like, "I'm a Yank and really, really do want an automatic."
Be aware that the country roads in the Cotswolds (which you will want to drive on to see some of the sights) will seem very narrow to you, so be cautious, especially when passing an on-coming wider vehicle such as a truck (I mean, lorry).
Happy Travels! It will be an adventure.