Driving 12 passenger van in UK

In the UK, vehicles of this type are known as minibuses; passenger vans or small buses which seat up to 16 people.

Generally you are allowed to drive a minibus in the UK on a regular driving licence as long as you are not paid for driving it, nor are your passengers charged, and it is "used for social purposes by a non-commercial body". There are some other minor restrictions listed which probably don't apply to you, such as health standards for drivers over 70 years.

A regular class C driving license in the US allows you to drive these vehicles with similar restrictions (that may vary slightly by state). And generally your foreign driving license should be accepted in the UK.

If you're using the minibus solely for your family, you ought not to have a problem with hiring it or driving it legally. I do recommend you obtain an International Driving Permit before you go, even if you wouldn't normally need it in the UK, because the IDP shows your driver license class using the same classes as are used in the UK and Europe, which may make it easier for the hire car company to understand that you should be able to drive a minibus.

Get an international drivers permit in the US, either from AAA or other agency, and try to ensure that they place Class C and D, D1 as classes of vehicle licensed to drive if your US Driving license does not restrict you to motorbikes or automatic vehicles with no more than 8 passengers.

Some states, like Florida currently where I reside, there is no distinction what private vehicle I may drive, just excluded from driving a Motorbike as I did not take that test, or a commercial vehicle or city bus. However, I did get a UK license when I lived there, and before 1997, so it does not prevent me from driving the newer classes of vehicle that were introduced and required further skill testing to drive in the UK, so that if I presented my old UK license as well as an International Drivers permit notated with the additional classes of small van and passenger vans up to 16 passengers, I should both be able to rent and then also drive in the UK in a planned upcoming family trip for non-UK family members.

Just be careful in the cities, towns and country villages, because the roads were not built for large or long motorized vehicles and making a turn or parking in city centers may be more challenging than you are used to !

If I understand the question correctly, you are asking about the possibility of obtaining a D driving licence for buses with more than 8 passenger seats in the UK as a US citizen. Setting aside the fact that D and C licences are arguably the most difficult permits to obtain in terms of practical tests and training time required, it would seem that you won't be able to train and obtain a driving licence in the Great Britain if you're not a Great Britain resident. Indeed the first step required to upgrade your licence is to covert it into a UK one, and this can't be done unless you reside in the UK:

You can't exchange your driving licence because you're not a resident of Great Britain.

Assuming you are in fact a GB resident, then the feeling I'm getting is that the whole procedure might last longer than your holiday. You need to factor in the bureaucracy required for the licence change, the one required to receive the professional entitlement, the training and examination time.