OK, having tried everything, and no reply from Blizzard on a ticket, we went for a complete re-install of windows from scratch and reinstall of Diablo 3 from scratch.
We even skipped the nice 3D graphics install and everything else - just totally bog standard windows install and then Diablo 3.
And no, it is being an arse. Error 315300 wrong login.
To be clear, we have tried with and without an authenticator set. If I put the wrong username or password it says no right away, but with them right it goes on to ask for the authenticator. That means it is checking they are right for both username and password. If I don't have an authenticator it does not ask or work.
We tried from another PC, and that works? Well, actually, if we try from my machine first it says no as another PC is logged in. So clearly it is accepting the details and giving a bogus error. So this is clearly connected to the Internet and checking the details. It is just broken.
Clearly cannot code the simplest of "Login" processes if their livelihood depended on it. WTF are Blizzard playing at?
This is proper fraud if you ask me - take my money and give me crap.