Does Scotland and/or England require an international driving permit?

It's not likely since the IDP is primarily a translation from English into another language (French?).

Thanks - that makes sense

We've been told by AAA that it is not needed, but nice to have as an additional form of ID, for example in case your US driver's license gets lost or stolen. For car rentals in France, England, and Scotland, we were never asked to show an international permit, just our US driver's licenses.

Unless you want to really annoy people, remember the country is the 'United Kingdom' (that is the short name). England and Scotland are part of the UK, they are called 'countries' but are legally similar to US states. Driving licence conditions are set by the national government.
The UK does not require an International Driving Permit for holders of US licences. Not because the licences are in English, but because they choose not to. AFAIK France DOES require an IDP for US licence holders.
The purpose of an IDP, is not just an translation, it is a certification of the validity of your licence. A policeman cannot be expected to know how to identify the licences for every country in the world. Whether the rental company asks to see the IDP is irrelevant, it is what the law says that matters.

I do not believe that you are required to have an international driving permit. I did get one and found that the information booklet that came with it was very helpful. Many road signs are not the same as in the US and can be confusing, as if driving on the left, shifting gears with an unfamiliar arm, traffic circles etc are not enough