First of all, rental cars are called 'hire cars' in England. Definitely best to make your plans in advance of the trip - if only not to have to wait in lines.
We chose Enterprise as we had both had good experiences with them on domestic trips and they will pick you up and drop you off at the airport, the hotel, or the train station - wherever you need within reason.
You will need an International Driving Permit - which is just a matter of applying for it online or at your local AAA office and paying $15.00. It's not a text, no pictures required. Enterprise did ask to see it though. They also wanted a copy of our accommodation information which I don't recall having to provide before, so bring extra copies of confirmation documents.
Also, before we left we printed out directions in and out of the major cities and where we needed to drop the car which was really helpful. It's no fun getting lost when you're trying to catch a plan.
After landing in London we took a train to Bath, and picked up the car the following day which was great as I really didn't want to have to deal with when we were jet-lagged.
We had read some good advice about driving in England where everything is reversed. The passenger needs to be a super-navigator and remind the driver to hug the curb when making a left turn, which way to pull out when backing up, and clear direction with the roundabouts. The roundabouts are actually quite fun once you get the hang of it, but you need to know where you're going.or it can be frustrating. Getting our of Bath and Cirencester was a nightmare as it was early in our trip and we didn't know the area so some of the signage was confusing.
A good map is indispensable and you can pick them up at Heathrow or the train stations. Don't show up at the rental car company without one.
The driving itself in the Cotswolds is quite fun which is good because you can't enjoy the landscape too much if you're focused on the driving. The roads are in great condition, like glass; there are lots of curves and hills, and you won't really find yourself on dull straightaways if you take the secondary roads.
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