Having checked with a ex traffic officer the basics are;
If the vehicle is SORN and you tax online you must wait for your new tax disc before the vehicle can go on the highway.
If your vehicle is currently taxed, and you renew online you have 5 days grace in which to display your new tax disc after the old disc expires (but you must still display the old one).
If you renew at the post office you will have a tax disc in your hand so no problems.
Failure to display a valid tax disc outside of the grace period would normally be delt with by the officer who stops you by means of a fixed penalty notice and £30 fine (this fine can vary by police force but most are £30).
As perviously mentioned ANPR camera will show the car as taxed so unless they have another reason to pull you, you will not get stopped for no tax, but when you have stopped or parked for example an officer can see you have no vaild disc and can easily nick you.
If you tax disc is stolen, and you have applied for a duplicate disc online, you can drive for 5 days from advising the DVLA without a disc provided you display the following infomation in the passenger side of your windscreen,
Road Fund Licence Stolen
Crime Reference Number #######
Hope this clarifys things for you
Traffic lights in London are just rough guidelines