Whilst the appeal time against a sentence is 21 days, there is one other route that you can use, and this might well succeed, it is this:
The Magistrates’ Court has a power to vary or rescind its own sentences if it considers it to be in the interests of justice to do so. This power is contained in section 142(1) of the Magistrates’ Courts Act 1980 (see Further Information below). This provision is often used to rectify mistakes where both the prosecution and the defence agree that an error has occurred, such as the court passing a sentence or making an order which it does not have power to make, or where it is agreed that the court has proceeded on the basis of an incorrect factual assumption. In such circumstances an application can be made to the court under s.142(1) to put the matter right. In exceptional circumstances the court can use this section to increase sentence, such as where it considers it has been misled by the defence.
It has to be done as soon as practicable, and in your case that would be shortly after receiving information from DVLA where your ban was more than you believed it to be.
You would have to get a solicitor to do this. The fact that they gave you 8 months more than the guidelines, and there is no note on the file to say why the Magistrates came to that conclusion, supports your claim that there has been a mistake, coupled with your WhatsApp post about 20 months,
The only other suggestion I have is that you will see that the above opportunity is available where the prosecution agree that a mistake has been made. I am not sure that they will help at this stage, but it could be worth asking the CPS to look at their file and see what they recorded in their paperwork.
I would find it strange that they also recorded the disqualification length incorrectly. If they concur with your recollection that you had a 20 months ban, that would certainly give you cause to have the case re opened…. Or even persuade the court to consider amending their own file straight away. (but that would not be too likely without a further court case for them to amend it. The advantage of that is that it is a re hearing in the Magistrates court, and it could be done by yourself (I would help with the wording)