can a vehicle tax exemption be back dated?

wilko said:

When you apply for your car tax using the excemption form the DVLA will normally refund any car tax refund from the award date. For an example, your car tax was due in April 2018 your award is dated July 2018 and you realised you get full or half rate discount then when you renew your car tax or you phone DVLA you will or should be refunded from April 2019 to July 2018 the date of your award. But to be 100% sure phone DVLA who will help you through the process.

What you are saying is that the log book would be amended not from the date of application but from an earlier date being when the exemption applied? When I had my own car from 1995 to 2011 I had a valid DLA High Mobility award from 1995 to 2004. I knew nothing about this exemption so continued to tax it as normal. Thankfully this avoided a problem when I didn't renew the DLA award in 2004 until 2011.

I doubt it because if that was the case you could have claims for repayment going back years. I do believe that when the application to have the log book changed to the disabled class that is when the car becomes exempt.
Any tax paid as a result of the car being in the PLG class even if you can prove that it should never have been paid is lost money.

Yes check this up with the DVLA - they are the authority on this.