London Heathrow to Cotswolds

Consider Cheltenham for your car rental - there's an Enterprise and Hertz

To comment on what you wrote here:
"- Take Heathrow Express (will purchase tickets 90 days in advance) to Paddington Station.
- Train from Paddington Station to Oxford, Banbury, or Moreton-in-Marsh
- Rent a car in Oxford, Banbury, or some other village"

Make it easy on yourself.
Coach from Heathrow to Oxford.
Rent car @ Oxford.

Heathrow has two coach stations. The one we use most often is down a hallway after we land at T5. You walk up to the window and buy a ticket. Less expensive by far than the train for tickets bought on the day of travel. Coaches (buses) leave several times a day for Oxford.
The coaches are air conditioned, have comfy seating--more comfortable than on most planes--and WiFi on board.
The coaches are brand new and very clean.
There's really no need to go to the trouble and expense of going into London and then back out again.

Oxford is 44.3 miles northwest of Heathrow airport.
(London is to the east of Heathrow.)

https://www.google.com/maps/dir/Heathrow+Airport+(LHR),+Longford+TW6,+United+Kingdom/Oxford,+UK/@51.6153804,-1.4164156,9z/data=!3m1!4b1!4m14!4m13!1m5!1m1!1s0x48767234cdc56de9:0x8fe7535543f64167!2m2!1d-0.4542955!2d51.4700223!1m5!1m1!1s0x48713380adc41faf:0xc820dba8cb547402!2m2!1d-1.2577263!2d51.7520209!3e0

You are adding miles to your trip by going into London and back out.
Distance from Heathrow airport to London: 15.7 miles.
Distance from London to Oxford: 56.1 miles.

You can now also use the Elizabeth Line to Paddington.

What rental car agencies are you looking at? Hertz, Avis, etc? They all have websites that will confirm opening and closing times. I’ve booked and used Hertz in Oxford more than once.

I’ve also arrived at LHR and taken a shuttle on the Northern Perimeter Road to Hertz and driven from there to Lyme Regis.

Use Google maps and type in
Naughton, Ipswich IP7 7BT, UK nearby walks

Many will pop up but you’ll definitely need a rental car to get to trail heads.

Try the Angel Pub in Buford or The Royal Oak in Winchcombe or The Howards Arms in Ilmington look like gastro pubs.

Why don’t you rent the car from Heathrow? It seems silly to spend all that travelling about and then pick up the car. Heathrow is right on the M4 Motorway so you don’t have to think about driving in London if that was a concern. Heathrow to Naunton is only 1.5hrs drive so it won’t make a lot of difference.

Please don’t get off a long transatlantic flight then rent a car the same day when you are jet lagged. Particularly don’t get off a long flight, take several trains, become even more tired then hire a car.

You will need to be back at Heathrow at 9.30 latest or 9 to return the car. I would stay at Heathrow the previous night.

Naunton is a lovely tiny village on the Windrush near Guiting Power (another lovely village I have stayed in, with a village cricket team and a cricket ground at which to play on weekends) and west of Stow on the Wold.

I'd get the car in Oxford and go via Burford. I wouldn't do it after an international flight. I fell asleep in a pub carpark doing just that not a million miles away, near Burford. Not a good experience.

After reading the posts of Jennifer and Nigel, might I suggest you take the coach directly to Oxford, then check into an Oxford hotel for one night?
You might enjoy having an afternoon of walking and sightseeing in Oxford, after dropping off your bags at your hotel. Get a good nights' sleep, and then head out the next day in your rental car.

This is the details for the Heathrow to Oxford Coach, scroll down to the bottom of the page to see about the fares, and how to book ahead if you choose to.

https://www.oxfordbus.co.uk/airline

Rebecca as always talks very good sense. Don't drive immediately off an Transatlantic flight and have a day to recover from jet lag. Oxford is a lovely town to explore on foot. There is plenty of choice for car hire there too.

Also drop the car off the day before you are due to fly out of Heathrow and book a night in a hotel on or near the airport.

Many pubs in the area you are stopping are owned by the local Donnington brewery including the Black Horse at Naunton
I can't comment if any of these are considered gastro pubs
see www.donnington-brewery.com

"Many pubs in the area you are stopping are owned by the local Donnington brewery"

Try a pint of Cotswold Gold produced by Donnington Brewery if you have a chance.

I'm of the opinion that you should rent the car at Heathrow and drive to Naunton. According to Google Maps, it is 80 miles, you can do that in 90 minutes. You know yourself better than anyone on here, so no one can really tell you whether or not you should drive after coming off the flight. You will be landing in the morning in the daylight, you will have a second wind from getting off the plane and being in a new country. Grab a coffee at the airport before you head to the rental agency and you'll be fine.

As another commenter above mentioned, you are not driving in London or anywhere near it. You will depart the airport and almost immediately get onto the motorway. For example, if you were to rent through Enterprise, you would exit their lot, proceed through one traffic circle, and merge onto the motorway and stay on that for almost the entire rest of your journey. (you can double check the route in Google maps)

Getting off the plane, hauling your luggage to the Heathrow Express only to have to haul across another train station onto another train, and then lug it to a car rental agency after that sounds like a chore, and more tiring than anything else. How do you plan to get from the train station to the car rental office? Walk? Uber? Black cab? More time and expense. Then you will need to do all of this in reverse order at the end of your trip, likely early in the morning depending on what time your flight departs - leading to possible unnecessary stress if any one of the connections is running late or super crowded.

I am speaking from experience. I flew to Heathrow in April 2019 with my wife and my kids aged 3 and 8 and drove to the Cotswolds. We picked up a van at the Enterprise Rent a Car at Heathrow and were on our way in no time. We were on the motorway pretty quickly, and it was a relief as it gave me space and time to get used to driving on the other side, and having the various controls in the car on the other side too. It was really no problem, and it was much more comfortable for everyone to be in our own vehicle right after the flight.

That all being said, if you insist on NOT renting the car at Heathrow, I would suggest the alternate option of getting direct from the airport to whatever town you plan to rent the car in, and avoiding going backwards into Paddington.

I should add, the people who are telling you to wait a day to recover from jet lag - what do you do the next day when you wake up at 4am because of the jet lag? Now you have to wait all morning for the rental place to open, and you have wasted 1 or possibly 2 days.

You get to your final destination, when you wake up at 4am with jet leg, then at least you can hang out in your own place and not worry about having to pack up already. Or you can step out and take a walk at sunrise and get some fresh air and explore your local area...or you can just relax and plan to get an early start on sightseeing.

Obviously if you are elderly, take all of this with a grain of salt.

I think you will struggle to get from Heathrow to Naunton in under 2 hours.
Will take much longer if you hit the Oxford bypass in the rush hour