Taken from one of the police forces websites (Tayside) but this is National afaik. Important bit highlighted in bold.
Have I got 14 days grace to get my new tax disc when it runs out?
No, there is no period of grace for road tax. If your car is on a public road without a valid tax disc you commit an offence and you could be fined.
However when you apply for a tax disc online or by using the DVLA phone service you are able to legally continue to drive your vehicle for up to 5 days after the tax disc has expired while you wait for the new tax disc to arrive. This will only apply when you have applied for a new tax disc before the current one expires. You should continue to display the expired disc on your vehicle, until the new one arrives.
You have advance warning when the tax is due to expire, either six months or twelve months. You should make arrangements to re-tax your car within plenty of time of the expiration date of your current tax disc. You can tax your vehicle on line, see website in related information for more details.
Irrespective of whether a car is used on a road the DVLA send out automatic penalty notices to anyone who has not renewed their tax disc (unless they have completed a Statutory Off Road Notice). However, the DVLA delay sending out such notices until 14 days after the licence is expired, which is where the myth that there is 14 days grace probably began.