Car Tax After Successful Appeal

The other day, the Tribunal awarded me 12 points for Mobility, overturning the DWP's ruling that I was not eligible for the Mobility component. The DWP's ruling came into force on 7 March 2018 so I am expecting to receive payments back-dated to that date.
However, I have now had to pay car tax, twice, in the period since 7 March 2018.
Questions:
Do I get my car tax back?
If so, how?
Or does ATOS' incompetence mean that I am out of pocket for having had to pay car tax?
Than you for any replies.
Sincerely, IanR

Once dvla have proof I know that they will refund whart you are entitled to.phone them and explain they can actually be helpful.

Just a follow up...

To explain briefly - after the DWP dropped me from full mobility DLA to nil points PIP, I had to pay full car tax. One year later, I paid full car tax again. Half way through this second year the Appeals Tribunal reversed the decision and awarded me 12 points mobility. Shortly afterwards, the DWP sent me a cheque for the allowance I had not received and a certificate which showed that I was entitled to "free" car tax. The procedure for getting reimbursed was as follows :-

1) I took the certificate and V5 (log book) and the certificate to the post office and "re-registered" my car as a disability vehicle.
2) Shortly afterwards, I received a cheque from the DVLA for the REMAINDER of this (second) year's tax.
3) I then wrote to the DVLA enclosing a copy of the DWP's certificate and explaining that I wanted a refund for a) the first portion of this year's tax and b) the whole amount of the first year's tax.
4) The DVLA have just sent me a cheque to refund the whole lot.

I am back where I started before the thickos at ATOS thought they would be clever and take away my entitlement. After nearly 2 years of stress it is finally over.

Ian