Usually the driver not regustered keeper is responsible to ensure car taxed, mot and insured.
Therefore, why is the learner not being prosecuted?
Wow! that would be a tough lesson. A learner drive (even a mature one) would be considered fair to assume a driving instructor's car would be registered and MOT'd... 
If that were the case then I better start asking to see the rego papers on the cars I rent!
We had a situation at Goodyear back in the early 2000s where they moved all the admin to Melbourne. All our cars were registered on the same day each year. Some sort of company arrangement with the DMR. Anyway, somehow I became aware probably from the date on the rego label, that our cars hadn't been reregistered. I parked my car in the car park and advised everyone else. Took Melbourne three days to sort it out and we did a block hire of vehicles from the local hire car company.
Some admin arsehole in Melbourne even suggested​ I should have shut up about it.
I reported her.
Can you imagine the crap if one of our cars had been involved in an accident even if it wasn't our fault?
