I would like to travel between the Cotswolds and Cornwall witout a car? Advice?

I can tell you my experience. In 2019 we spent 8 days on a walking tour of the Cotswolds. At the end of the tour we had about 4 unscheduled days and we randomly decided to take a train Cheltenham (technically still the Cotswolds I think) and do some day trips from there. One day we got on a train and thought we were headed to Cornwall but we got on the wrong train in Exeter and ended up in a little place in Devon called Dawlish. The next day we took a train to Cardiff and back. So we got a taste of both Southwest England and Wales. In 19 days we are headed back over to the British Isles and will be spending 3 days each in Cornwall and Wales as well as 5 days in the Cotswolds and 3 days in Bath. Both trips entirely without a car. We inadvertently ended up taking the tasting menu last time and this time we are going back for more.

It's going to take at least half a day depending how far south in Cornwall, almost a full day if you go down to Penzance.

Many of the well known gardens are accessible by bus (more than many people might think), although in a number of cases a degree of patience is required to do so.

I'm not sure how you are getting to Moreton-in-Marsh (if that is where you are staying) to start your vacation but to me there are two sensible ways to get to Cornwall from Moreton,
a) train to Reading, change, train to Plymouth or Penzance;
b) Pulhams bus 801 to Cheltenham Spa, local bus CS Royal Wells Bus Station to Railway Station, train CS to Plymouth (journey time 3 hours), change at Plymouth for stations to Penzance.

The bus is the longer route by about 1 to 1 1/2 hours, but arguably the more scenic.

EDIT- The stretch of railway through Dawlish is one of the most spectacular sections of coastal railway in the UK. It was totally washed away in a severe winter storm several years ago and had to be totally rebuilt. In a normal winter storm it can be pretty spectacular with the breaking seas.

6 days in The Cotswolds + 5 days in London = 11 days yet you say that you will be in the UK for 14 days. It is not worth going all the way to Cornwall for 3 days being as it takes the best part of a day to get there - and another day to return to London. Forget travelling to France. You need to consider somewhere not so far away.

You don’t say where you are staying in The Cotswolds - it covers quite an area.

Not sure yetwhere we are staying in the Cotswolds. We could add a day or two more to our trip.
I agree Cornwall is a long way. When I look at alternatives, Dorset? travel time looks long as well.
I am open to suggestion. Not really interested in top tourist sites love natural beauty, gardens, and history.

Dorset is not too bad for travel time. Get to Bristol then the direct train to Weymouth. From there you've got quite reasonable transit along the Jurassic Coast, and also to the Isle of Portland.

I would suggest that perhaps 6 days is too many for The Cotswolds and that to make it more viable to go to another region, you could divide the time more equally.

Direct trains also go from Cheltenham to Exeter, Plymouth and into Cornwall - although in some cases you may need to change trains.

Direct trains also go from Cheltenham to Cardiff taking about 1 hour 15 minutes.

The nearest rail station to the best Cotswold villages with direct services from London (Paddington) via Oxford is Moreton-in-Marsh. Trains on this line continue westward to Malvern and Hereford.

It all depends on where you are staying in The Cotswolds as to which are the most appropriate rail routes.

If in the north Cotswolds, don’t forget that Oxford and Blenheim Palace are not far away as is Stratford-upon-Avon. You should also visit Bath - if not already done so.

Pullhams is one of the main companies in the north Cotswolds.

Trains from Cheltenham Spa to Weymouth are not direct. You always have to change at Bristol Parkway.
To reach Cheltenham Spa from Moreton in Marsh you have to change at Worcestershire Parkway so it is a 4 train route, hence the suggestion of a bus to Cheltenham.
The normal route from Moreton to Weymouth is actually change at Reading and Castle Cary, or change at Reading and Southampton.

Yep - I’m wrong concerning direct trains from Cheltenham to Dorchester/Weymouth. I think that GWR have changed their services since I last checked this route. (Now corrected in my post).

Have you researched how you'll get around the Cotswolds from your base--assuming you don't intend to make that base the starting and ending point of daily hikes?

I believe bus service in the Cotswolds is pretty infrequent. I fear you'll run out of easy trips via public transportation before six days is up. And I would question that duration of stay unless you plan a lot of hiking. It's a beautiful area and there are many cute villages, but in terms of sightseeing beyond rural hikes, I think it could get repetitive rather quickly.

I took a one-day small-group tour of the Cotswolds from Moreton-in Marsh a few years ago. Although I'd have liked most stops to be a bit longer, I otherwise was happy with what I saw. Given that I can't do rural hikes, I didn't feel I had other practical options.

Should we make it to Cornwall we would stay in one place and see what ever is close.
Is there a direct train from Bristol to Plymouth? Anyone know?

Read the official Tourist Board website for Cornwall.

Yes, trains run from Bristol (Temple Meads is the downtown station) to Cornwall with some going all the way to Penzance. Looe is one seaside town that can be reached by train via a change. St.Ives is another - change at St.Erth. Penzance is not particularly attractive.
Fowey is also a nice estuary town in Cornwall but you would need a bus connection from the railway.

www.nationalrail.co.uk
(Book ahead specific train for cheapest tickets. Some journeys are cheaper by splitting ticket at a calling place).

Fowey is accessed from St Austell Station where buses interchange with trains. Two buses an hour. Or alternatively by ferry. For the ferry bus from Liskeard to Looe or Polperro, change there.
At Fowey buses can not physically reach the harbour. They terminate at a pub at the top of the village from where it is a steep walk down (and back up) to the harbour and village centre. Opposite the stop there is a book exchange in an old phone box.
Looe is split into East and West Looe. The station is in East Looe, which is the more picturesque side of town.