"UK road tax is mandatory for all car owners, even those who are tax-exempt. When you’re selling your car, there are a few important steps to take to comply with road tax regulations. The current registered keeper (you) is responsible for the road tax until the moment of sale, at which point responsibility passes to the new owner. Historically, road tax was transferable from one owner to the next. Since October 2014, the UK government changed the rules and made road tax non-transferrable. This means that you, as the seller, cannot transfer your road tax to the new owner when selling your car."
When you notify the DVLA you've sold your car, they will automatically know to issue you a refund for the road tax but as it is Historic it will be £0.
You shouldn't notify the DVLA at any point before the ownership has been officially transferred. Doing so could result in an unwanted fine — it's illegal to drive, park or keep a vehicle on the road without road tax.
Also if selling privately you will need tax in place for any test drives with potential buyers.
Some useful info here:-
https://www.car.co.u...-complete-guide