You are able to buy the tax disc from the post office for up to 14 days after the expiration of the previous one. After that time they cannot issue you with one as they don't hold them any more (as told to me by a PO employee)
You do not have 14 days to tax a car, but 14 days is around the sort of length of time past due, when the authorities will start to take action, it should not be taken for granted, it could be less.
It is not an automatic fine. I have been late twice in renewing mine (once I was on holidays and once I lost the letter due to moving house) and I wasn't fined either time, so it is definitely not an automatic fine
my friend was due her mot yesterday and unfortunately the computers had gone down so she couldnt get it done until a later date but she was planning on the car passing the test and then going to get her tax that had expired on 31st march after not getting her mot done she rang the mot centre and asked would it be possible to get an exemption certificate so that she can tax her car and she got put thru to the manager who then told her she had 14 days to tax her car so not to be worrying about it.