You said, "We could return to London on Thursday or find somewhere else in the region to visit for another day (Stratford-upon-Avon or Warwick or ???)"
You are on the right track there. Both Stratford-Upon-Avon and Warwick are great places to visit. Stratford-Upon-Avon of course has all the buildings associated with Shakespeare; New Place, Anne Hathaway's Cottage, Trinity Church where Shakespeare is buried, etc. It has the Royal Shakespeare Theatre, where you can see a play if you are there the right time of year. Stratford-Upon-Avon is also a great place to take a long stroll, walk by the river, or have a good lunch in a pub. I was just there in May, and wished I could have spent 2 or 3 more days there.
If I were you, I would also go to Warwick. Warwick Castle is amazing. The town of Warwick near the castle is also interesting.
We did not have a car during our visit in May. But both Stratford-Upon-Avon and Warwick are on the train line from Oxford.
We stayed in Stratford-Upon-Avon, and took a small mini bus tour from there to the Cotswolds. This ended up being a really efficient and hassle-free way to see the Cotswolds. With someone else doing the driving, we got to see more than we could have done ourselves with a car.
The company we used was the "Go Cotswolds" Tour, hosted by Tom. We were picked up at our hotel in the morning by Tom and his mini-bus.
We went to Chipping Campden, Broadway Tower, Stow-On-The-Wold, Bourton-On-The-Water, and the Rollright Stones, to name a few of the stops. Tom gave out walking maps in some of the towns, and made recommendations for shops selling coffee and cakes, or lunch. Plenty of time was given for us to walk around and take photos at each stop. Tom runs tours of the Cotswolds on Wednesdays, Fridays and Sundays. ("Go Cotswolds" Tour is highly rated on TripAdvisor, and has just received the Trip Advisor Certificate of Excellence for 2016.)
http://www.gocotswolds.co.uk/
http://www.gocotswolds.co.uk/tour-details/
tom@gocotswolds.co.uk.
I would give some thought to either 1. contacting Tom and making arrangements for him to pick you up at your Stow hotel, OR
2. changing your hotel reservation to Stratford-Upon-Avon and going from there. This is the best way to see the Cotswolds without a car, in my opinion.
If you have a car, and prefer to do that way, that's great. But we did not want the hassle of reserving the car, picking up the car, finding our way around the countryside just for one day tour of the Cotswolds. If we had been staying in the Cotswolds for a week, that would have been different. We would have probably rented a car. For our needs, the mini bus tour plus the rail line going through Stratford-Upon-Avon and Warwick was the easiest and most hassle-free (i.e. relaxing) for us.
FYI--We saw several car rental places in Stratford-Upon-Avon.
