Practical Answer? ...
They will not pester you for ages. In a couple of weeks, you will get a "Reminder".
As long as you re-tax within 1 month, you can go online and use the code on the Road Tax Renewal form, which should have been sent to you already.
If you don't retax with a month, then you have to phone your local DVLA office (NOT Swansea), and pay the arrears for the month you missed, and then go to the Post Office and buy a new disc which will be dated to the start of the month in which you pay.
Net effect ... you don't get away with any gaps in your road tax.
As a consequence of which ... you won't be pestered too much.
If the Police pull you up, they will just tell you to "Make sure you sort it out, Madam".
And yeah yeah ... I've been there (2 1/2 month late)