Speaking personally about a generic query such as this (ie not OcUK specific)
Where is the upside?
100% of people can see it on v4
1-2% of those 100% could see it on v6 too..
Where is the business case for the tens of thousands of £ it would cost to implement?
A very good question, and the first one any business asks when they start looking at this.
Let's start with the business case for the access provider (BT, Virgin, etc.). It is no longer possible to obtain any meaningful number of IPv4 addresses from RIPE so they are left with what they have or what they can buy from others. Regardless, there is a finite number available and eventually they *will* completely run out. At that point, if they wish to continue selling broadband, they need to do something.
Even IPv6 doesn't completely solve this problem, many websites (Such as OCUK) are IPv4 only so they have to provide access if they want to keep their customers.The answer is some kind of transition technology, typically CGN (Carrier Grade NAT) - exactly the same NAT your home router does however at the huge scale large service providers operate at. This allows SP's to give their customers private IP's and let them share a public address with others. The problem (Aside from numerous technical challenges and UX problems) is that it's expensive. That fancy CGN hardware does not come for free and it's a completely new expense that never existed before.
IPv6 helps here. If they give their customers an IPv6 address as well as CGN, all of the most popular content can be delivered native v6, completely avoiding the nasty, expensive, CGN. That's terabits and terabits of data. Of course, as more of the internet supports IPv6, less and less traffic will have to be CGN'd.
For a website provider, the main driver is going to be UX (User Experience). SP's aren't doing CGN yet, but when they do, it's another hop between the content and the provider. As it's expensive, and most popular content is already native v6, it remains to be seen how oversubscribed the CGN boxes will be.
For webhosts that aren't doing v6, they are going to be in a bit of trouble. It's not easy to get ready for v6 (As you've said, it *will* cost thousands) and it takes time, time they may not have if the websites they host start demanding v6. Amazon AWS as well as many other big hosts offer (Or are close to) v6 now so it's not too bigger stretch to imagine customers migrating in order to continue to provide a great UX.
In terms of OCUK specifically, they won't gain anything immediately by moving (Other than appearing forward thinking) but they will gain in the coming years as the exhaustion of IPv4 starts to bite.
Phew, long post. I kept that as short as possible but it's hard to explain the true drivers without a wider view of the internet.