Have a look at section 75 of the Road Traffic Act.
It is illegal to sell on unroadworthy vehicle. That doesn’t apply just to traders, it says “no person” so you do have recourse against the original seller although you have to remember that the original seller may not be aware. However that would be my first port of call.
Otherwise you are correct that a private sale when there is no description as to the condition, does not have to be of satisfactory quality or fit for purpose, it just has to be as described.
There is actually no privity of contract between you and the MOT tester although you could possibly rely on the neighbour principle established in the case of Donoghue v Stevenson which is well settled and well-established law: https://en.wikipedia.org/wiki/Donoghue_v_Stevenson
As well as referring the matter to DVSA (which you have already done and they will investigate the test station) you can also refer the Seller to Trading Standards although they may not get involved with a private seller. Nonetheless they should because it’s Trading Standards that prosecute breaches of section 75.
There is however no reason why you cannot refer this to the police although you are doing don’t do it on the telephone because you will be speaking to an amoeba and you will only get fobbed off that it’s a civil matter. Turn up at the local nick with a copy of the section of the act and tell them that you want the Seller prosecuted.
Armed with all those threats (before you actually do it) the seller may decide they will agree to refund you although be careful it is not seen as blackmail. Blackmail is the unwarranted demand with menaces and although you are making demands with menaces (to go to Plod and Trading Standards) I don’t think your demands are unwarranted.
You need to remember that if it has had a test and failed, the original test is no longer valid.
A lot comes down to whether you want to return it for refund or whether you want to get it repaired and take the Seller to court for the cost of the refund.
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