What Car Q&A - Can road tax be back-dated?
You only need to pay tax from the date of acquisition. It’s the responsibility of the previous owner to pay the car’s road tax or declare the car SORN (statutory off-road notice) up to the point they hands you the V5 registration document.
However, the moment you become the new keeper, it becomes your responsibility. That means you must tax the car, even before driving it home.