The buyer is protected by the law of contract and also has come-back if the car turns out to be unroadworthy. But he is not protected by any consumer legislation as purchasers are when buying from a dealer.

Your problem might be insurance because policies no longer allow policyholders to drive other cars not owned by them. Your best way out might be www.sure-sell.info

If you can wait until January when the market traditionally picks up you should do a lot better than trying to sell in December.