The Cotswalds

We took the train from London Paddington to Moreton-in-Marsh then the bus to Stow-on-the-Wold where our B&B was. We hiked and used the bus system to get around the area although you could just use the bus.

You can also take the bus from Heathrow to Oxford and then pick up the train to Moreton in Marsh in Oxford, rather than take the train to London and then to MIM - we've done both and the bus-train combination may be faster.
In addition to local buses (which are fun but unreliable, in our experience) and walks, you can try local taxi services. The distances between villages and towns aren't far.

Even if you were travelling from Heathrow the bus to Oxford and then train is not the most elegant solution because train and bus stations in Oxford are nowhere near each other, it must be about a 15 minute walk between them.

It will be much better with a rental car.

Take the train directly to Moreton in Marsh and use the buses from there. We used an occasional private driver to get to places that weren't on the bus lines.

You can also take the bus from Heathrow to Oxford and then pick up the train to Moreton in Marsh in Oxford, rather than take the train to London and then to MIM - we've done both and the bus-train combination may be faster.

The train and bus stations in Oxford are nowhere near each other, it must be about a 15 minute walk between them.

Actually, it's a bit under 10 minutes between the train station and the Gloucester Green bus stop and it's an easy walk. I walked between the two and had no problems at all. Of course, if you're coming from London itself, that wouldn't work. But if you want to get to the Cotswolds directly from Heathrow, I think it's a viable option and saves you having to detour through London.

Of course, I would also recommend staying in Oxford for a few nights before you head to the Cotswolds, as it's a beautiful city and well worth the time to visit it. :-)

And, there's always walking. Lots of support for lots of paths.

https://www.rabbies.com/en-us/tour/oxford-traditional-cotswold-villages-day-tour-from-london

Lots of options here for going without a car:

https://community.ricksteves.com/travel-forum/england/day-tour-of-the-cotswolds

If you take one of the tours from Moreton-In-Marsh, I recommend staying there the night before. It's a lovely Cotswolds village. Or possibly stay in Oxford the night before the tour and the night after. Lots to do in Oxford.

It's Cotswolds BTW.

Bus which isn’t on a regular schedule or cab.