Cotswolds base without a car

The easiest place to get to in the Cotswolds without a car is Moreton-in-Marsh which you can get to from London Paddington station. Once you are there there are buses to nearby towns like Bourton-on-the-Water and Stow-on-the Wold which are both pretty well connected for buses- so any of those could be a good base. I once did a very nice walk from Bourton to Stow via the Slaughters which are a pair of very charming typical Cotswolds villages. One thing to note about many of these buses is you need to look up/know the timetables in advance as many don't run very frequently. I've found the info given by Google maps to be generally very accurate for this but you probably want to cross check with the bus website. I've also found that taxis need to be booked a few days in advance as local taxi companies can book up quickly.

Good luck! I'm sure others here will have additional suggestions.

We did the Cotswolds staying in Chipping Campden at the Volunteer Inn. It was excellent.
Get a rental car, much better than public transport.

Googling “Cotswolds without a car” brings up a number of websites with links to places to stay, timetables etc.

As has been said, the easiest place to start is Moreton-in-the-Marsh as this is directly accessible by train from London.

That said, if it were me I’d hire a car.

I stayed in Moreton in Marsh back in September. I booked a small cottage (Airbnb) on the high street and it was perfect for our group of three. 5 minute walk to the train station and grocery store. There are several pubs along the high street that serve food. We actually did a small 2 day tour that met us each morning at the station. We did the North Cotswolds one day and the South the next. It was wonderful. We also did a day trip into Oxford on our own. This was just a 20-30 minute train ride. There are many small group tours that you can google, The tour we did was: https://www.cotswoldsguidedtours.co.uk/. (I hope this is allowed to recommend). I highly recommend a tour if you do not have a car. We also decided on a private car from LHR to MIM but you can do it by train fairly easily.

Taking a tour on at least one day would be a way to see a bunch of places very efficiently. Tour participants don't get a great deal of time in any single village, but the villages tend to be very small, so a quick stop might be all your sister needs. I'd research the bus schedules carefully in advance so I could figure out where I wanted to go that wasn't covered by the tour. The public-bus network is a bit thin, and I suspect it would be difficult to see more than two little villages in one day--thus the practicality of a bus tour or two.

I trust your sister plans to do a good bit of hiking/walking, because I'd otherwise say 6-7 days would be a very, very long time in the Cotswolds. The area is lovely and the villages are cute, but at some point I think the villages would begin to look alike.

There are tours to the Cotswolds from both Bath and Moreton-in-Marsh. It is possible to sleep in Oxford (which has a lot of worthwhile sights) and catch the GoCotswolds tour departing from the Moreton-in-Marsh train station.