My friends forgot to renew their tax (good excuse, they were moving house and having a baby at the same time so had bigger things to worry about!) and had a letter (with fine) within 2 weeks.
The thing you have to contend with now (which I should have mentioned earlier, my apologies) is the Statutory Off-Road Notification scheme (commonly called SORN), which was introduced just a few years ago.
Upon expiry of the license, DVLA now expects the registered keeper of the vehicle to either renew the tax
or make a declaration that the vehicle is off the road. If you do neither, then the automatic fine system kicks in, although 2 weeks sounds a very short length of time. I know there have already been concerns about people being out of the country for a few weeks and not being able to make the declaration coming back to find fines in the mail.
Under the old regime everybody knew his responsibilities -- Renew the tax or take the car off the road, and if you get caught driving without a valid tax disc, you're in trouble. The system now effectively demands that one makes a statement of intent
not to do something, which I think is wrong.
I can understand the reasons for introducing SORN to an extent though. The 13-month year trick I mentioned is the case in point, where people knew they could probably get away with at least 7 to 14 days. If they happened to get stopped in that time, they'd just renew from the first of the month and nothing lost. After the first couple of weeks, they'd then perhaps be a little more cautious about driving or not use the car at all, then renew on the first of the following month, thus effectively getting a month's use (or at least a couple of weeks' use, for free).
Now everything is computerized the chances of getting stopped are much greater, since ANPR (Automatic Number Plate Recognition) can automatically flag up any passing car with expired tax.