Although a tax disc can be purchased in advance of the current one expiring, there is no grace period following expiry in which you can continue to use the vehicle. This applies even if you intend to "get around" to buying the next tax disc, which will start from the date the previous one expired. If your tax disc expires on a Sunday, in theory, your obligation is to buy the new disc before it expires and not wait until the following Monday. However, it would be an extremely harsh court that imposed a punishment in those circumstances.
http://www.motorlawyers.co.uk/offences/vehicle_tax.php