In reply to Bjartur i Sumarhus:
If the seller did the notification by post it takes much longer to update and the process as far as I know is still separate for the V5 and the tax refund (I've you can notify change of owner, but choose not to ask for a tax refund). Online it is all in one - the sold notification automatically triggers the tax refund.
The beta online notification can apparently can take up to five days to update the website database. Even then it is not that clear - I sold a car earlier this month and it does comes up at near the top of the webpage as untaxed with a "X untaxed" and in box coloured in red, yet further down the same page it clearly says "Taxed and due" whatever that actually means!
Read that there have been into the thousands of "honest" mistakes last year and suspect there will be many more. Last year I never received a reminder to renew the car tax, and there must be many that just forget because of this let alone the folk that buy a car thinking that the tax is still valid after purchase.