The poster who said that it used to be OK to use a BB if doing a job for the holder, is mistaken. Provided you are insured to do so, you can drive a Motability car to do shopping etc. for the registered owner, but a BB is solely for the use of the disabled person, and the disabled person must not be sitting in the car while someone else nips into the shops for them, they must get out and do it themselves, as any able bodied person should park, and walk to where they need to go.
I think that as long as your Nan tells the truth OP, that she didn't know about your cousin using the BB, and she had accidentally left it in his car, she might be OK, although it's likely to depend on who deals with the case, as she is legally responsible for the BB. For future reference, can I suggest that she carries the BB in her handbag, so that she can use it whoever she travels with, and won't risk someone else using it without her knowledge.
Your cousin however, is likely to get a £1,000 fine for misuse of the BB and failing to show it to a person who is entitled to see it.
Had he told the person who caught him that he had dropped his Nan off and was waiting for her, they would have insisted on staying with him until she reappeared, so he would have had to 'fess up eventually, and of course CCTV would have proved that she was never there in the first place. I fear that this is going to be a costly lesson for him OP!