flying in/out of Heathrow. Staying 5 days in Chipping Campden.

Train to Oxford, rent car there. Or rent car at Heathrow. NEVER rent in London

Have you booked Chipping Campden on your arrival day? Most people on this forum (myself included) wouldn't recommend driving over 2 hours on busy motorways, on the left hand side, on arrival day after a long haul flight. If you insist on driving that day, theres no need to go into London. You can rent a car from LHR. That is likely your best bet for an automatic transmission.

Or you could take a direct coach from LHR to Oxford. Pick up the rental car there. Better yet, spend the first night there and sightsee, and THEN pick up your rental car.

We spent 6 weeks in the UK last summer. Didn’t need car in London. Took train to Edinburgh for a week and rented a car when we left there to drive the coast. No problem driving. People were patient. But that was the countryside. No city driving. Left car in Belfast before heading back to the states. Will definitely rent a car for the Cotswolds. Just not sure of best place to do so. Thanks everyone for your help!

What Claudia said.
The trains go from London to Oxford, then some trains continue on to Stratford-upon-Avon.
Oxford and Stratford-Upon-Avon will both have car rental places.

What CJean said.
If you are arriving at Heathrow and going straight to Oxford, take the coach (bus) from Heathrow Airport to Oxford. It's easy to find the Heathrow coach stations.

"Rent car in London and drive to Campden or take train closer and then rent car?"
The worst of all possible options would be to rent in London. If you mean downtown, in central London.

Renting at Heathrow airport is well outside of central London and would be OK.
If you want specifically an automatic, you may find a larger pool of cars to choose from at Heathrow airport rental lots.

Do check what's available (if you want an automatic) at car rental lots in Oxford and Stratford-Upon-Avon.

Rebecca used to be correct- trains ran through to Stratford on Avon. Either this June or last December (don't remember which) that became a Leamington Spa to Stratford on Avon shuttle every 2 hours.

A single through train extends from Leamington Spa to London Marylebone at 2133 (no northbound equivalent) on Monday to Friday only.

I’d just rent the vehicle upon arrival at Heathrow and drive to Chipping Campden. As stated it will take about 2 hours, but most of it is highway driving. Time of day and construction on M40/A40 will affect the total time. Only you know if you can drive for 2 hours after a long flight.

isn31c, thanks for the updated information on that train/shuttle to SUA!

There is a 3 day tour option from London which my aunt did recently which is available from Cotswolds Tour Bus by Shakespeare Coaches. They pick up from Paddington Station and do a day in Oxford, a day in Stratford-Upon-Avon and a final day touring the Cotswolds before returning to London. Accommodation is included in Stratford-Upon-Avon in a Holiday Cottage. I would not recommend taking a car as the traffic can be difficult to negotiate if you're used to driving on the left, plus most satnavs don't really work in the Cotswolds due to signal issues. Stick to a tour and make the most of your time and stay safe!

I rented a car in London at the Victoria Station, as I was seduced by the low monthly price. I'd do it again for the right price, but it's certainly a stressful way to do things.