Two towns in the Cotswolds that do have good train connections are Moreton-In-Marsh and Stratford-Upon-Avon.
There is an excellent tour company that picks up passengers from both of these places.
Go Cotswolds tours is in a small Mercedes mini van, and goes to many of the scenic Cotswold villages.
https://www.gocotswolds.co.uk/tour-details/
Two years ago, we stayed in Stratford-Upon-Avon, and took this tour. Tom (the tour company owner) picked us up at our hotel.
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 was born and raised in Stratford-Upon-Avon, so he knows the Cotswolds quite well. We enjoyed his stories and explanations of local landmarks. Being a local, Tom can also suggest some great restaurants around Stratford-Upon-Avon.
http://www.gocotswolds.co.uk/
It is a great tour, with an excellent tour operator who reminds me of a very young Rick Steves.
You may want to rent a house in Moreton-In-Marsh or Stratford-Upon-Avon.
When guests come to stay, you could take them on the Go Cotswolds tour instead of driving all around in a car.
You might also stay in one of the Cotswold towns that begin the tour, say Chipping Campden.
Bus service does exist in the Cotswolds between towns. Not many per day. Some days no service. But if you have settled into a cottage, and are doing hikes during the day, you may not want to travel every day.
There are taxis in the Cotswolds. You could use bus plus short taxi ride to get you to castles, etc.
You can take the train from Moreton-In-Marsh to Oxford, Warwick (Castle), and Stratford-Upon-Avon.
You can take the train from London to Moreton-In-Marsh, Oxford, Warwick, and Stratford-Upon-Avon.