Blue Badge parking permits in England and Scotland

Not directly answering your question, at all, but I am planning to bring my 89 year old mother to Scotland. I've been perusing castle websites and they generally have an impressive amount of information on handicapped accessibility, that you might find helpful at some point. Also, caregivers get in free to some places. One place said that having a blue badge would be sufficient documentation. I contacted them and said I was from the U.S. and wouldn't have a blue badge. They told me to bring the handicapped card I hang from the rear view mirror.

The problem is we are traveling around the country, so which council do we apply too.

The councils that have local government authority in the area you are visiting.

For example this is the process applied by Cornwall County Council:

Using a non European Union disabled parking badge in Cornwall

If you want to use a disabled parking badge from a country outside the European Union, we need some information from you before you arrive in Cornwall.

This will help ensure you don’t receive a penalty charge notice.

Please email [email protected] at least a week before your visit. You will need to send us the information detailed below:

A copy of the disabled parking badge you will be using

Your vehicle registration number (we understand you might not be able to tell us this if you’re hiring a car)

Dates of your stay in Cornwall

A general idea of where you’ll be in Cornwall (for example, Padstow or Truro) – you don’t have to provide exact locations or a daily travel plan
When we confirm that you can use your badge you will then be able to park as detailed below.

There is information about eligibility amd applying here:
https://www.citizensadvice.org.uk/benefits/sick-or-disabled-people-and-carers/help-for-disabled-travellers1/blue-badge-scheme/applying-for-a-blue-badge/

As has previously been stated, these are intended for UK residents and the eligibility criteria are strict, also the amount of supporting evidence. My husband's application took four months to be approved.

As others have said the Blue Badge scheme is simply not designed for visitors to the UK.

You should also note that whilst the word "handicapped" is still in regular use in North America it definitely is not in the UK and you are likely to offend if you use it. Please use the word disabled instead.

FWIW, the term "handicapped" is not in regular use in the parts of North America I frequent.

Good to know about "handicapped" vs. disabled. I am a little surprised, my area tends to be quite progressive for the U.S. I did a google search and indeed the cards are referred to as "disability" card, but then the two words seemed to be used interchangeably in the text, and parking spots are labeled as handicapped. My mother's permit says handicapped. Sounds like we are in transition to more appropriate terminology.

Last month in Ireland, I just hung my handicap placard over the rear-view mirror and no one hassled me when I parked in handicap parking spaces. I use a cane so I can't speak about wheelchairs. Some of the B&B's we stayed in had a handicap spot right in front which worked out nicely.

I can't speak for Ireland, but in England (wales and Scotland may be different), there are two sorts of parking - private land and council/local authority parking.
In the latter, you must have a blue badge to use the disabled spaces. Otherwise you risk a fine.
Then there are the private car parks - mostly run by commercial companies who are there to make money (the word pirate has been used, among others). It is possible that you might get away with using a foreign "handicapped" placard, but most likely you will incur a penalty parking charge.