Any grace period on taxing a newly bought car?

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Unless there has been a recent change...

At the end of the month if you haven't got the new V5 you use the green slip you got from the V5 to tax the car online, and ensure you tick the box that says "I am the new keeper", fill in your address and this will generate a new V5 in your name as well as taxing it in your name.( and refunding the previous keeper if any full months tax remain).

If it won't accept the number from the green slip, then a new V5 has actually been issued, and you will have to wait for it to arrive.

(or the green slip you have isn't from the last V5 issued, but let's not go there)