Give some thought to this plan. From Heathrow you could take the bus/coach to Oxford. There is a bus station at Heathrow Airport from which many buses or coaches go out to various towns and cities in England. You can just walk up to the bus station window inside Heathrow and buy a ticket. Fairly inexpensive. At Oxford, catch a train to Stratford-Upon-Avon. You already mentioned that your wife wanted to see RSC or a show in Stratford-Upon-Avon.
A year ago, we took the train to SUA, stayed for three nights. One day we explored the Shakespeare sights including RSC.
The next day we took an all day tour of the Cotswolds with the company Go Cotswolds. It was great! We were relieved to not have the hassle of renting and returning a car.
Read a Trip Report I did of our tour with them here:
We found Stratford-Upon-Avon to be a good base for exploring the area and it also has some good pubs and restaurants. The Avon River at the center of town is beautiful with boats and swans on it. We walked north out of town on a path beside the canal and watched some narrowboats pass through some of the locks. My husband helped the boat owners with the locks. Very interesting to watch boats go through the locks and to chat with the owners who lived aboard.
Not far from SUA is Warwick Castle which would interest a 13 year old boy I am sure.
From Stratford-Upon-Avon, just catch the train to London.
If you would prefer to drive, there are companies that rent minivans. This might be good for a tall person.
Some friends rented a minivan from Parkers Rent A Car and were very pleased with the vehicle and the service. The company is based in Sussex, but will deliver a car or van to Heathrow. You can see their ad in the latest edition of Britain Magazine. Do mention that you want an automatic when reserving if that is what you want. Don't know what you prefer.
Hope this helps.
Have a great trip.