British passport holder travelling to Kuwait and will be driving to Dubai via Saudi. What visa will we require?

You will need a 3-day transit visa. At present, this is the only way for westerners to enter the Kingdom of Saudi Arabia as independent tourists. (Excluding a sub-18 hour air transit.) The visa must not be endorsed "Entry by air only".

Unfortunately, I don't think this category of visa is open to you unless you are a resident in one of the Gulf Cooperation Council countries; or can otherwise demonstrate a "need" to drive across Saudi Arabia.

Visa applications may be made here: https://enjazit.com.sa/Enjaz/PersonServices

I will add to this to say, the Kingdom is not a country that likes to admit foreigners, excluding their neighbours. You cannot simply enter the Kingdom without having a good reason for doing so.

You will not be admitted to KSA if your passport includes any evidence you have ever visited Israel. Any women in your party must be relatives or in the company of their husbands. Of course, they cannot drive a car.

See also https://www.gov.uk/foreign-travel-advice/saudi-arabia/entry-requirements

The only guy I know who has done this posted an extensive trip report about it on FlyerTalk. http://www.flyertalk.com/forum/trip-reports/1556689-road-trip-saudi-arabia-qatar.html

I believe he is a New Zealand citizen resident in the UAE.