Buying a used Car from a Private Seller is basically a matter of 'what you see is what you get', no Warranty no Guarantees.
However if there is information withheld about the condition of the Car, or a fact is stated which is untrue (such as "the Car has a current RWC"), then in that case the Seller is responsible for rectifying the situation.
It sounds like the Seller told you about the RWC rather than putting it in writing, so your problem will be proving what was said.
Basically you need proof that the Seller stated that the Car had a current RWC, either in an e-mail, SMS, recorded message (voice mail), or a Witness to the conversation.
If you have none of those then it comes down to the Sellers word against yours, in which case you may unfortunately have to cover the cost of the RWC yourself.