Unfortunately its very much caveat emptor in these scenarios.
From the CAI website:
I bought a second-hand car privately through a small ad in the newspaper. The ad claimed the car was in perfect condition, but now, two weeks later, it has broken down. Can I claim a refund because the product was not as advertised?
When you buy a car from a dealer or a garage, you have certain consumer rights that protect you if the car is not of merchantable quality, not as advertised, or not fit for its purpose. Unfortunately, if you buy from an individual who isn’t in the trade, the transaction is not a contract between a business and a consumer, but a deal between two individuals. This means you have very little legal protection if the car turns out to be faulty, as the purchase falls outside the scope of consumer law. Your only right to redress in this case is if the car was sold with a defect that would endanger those driving it. For these reasons, if you buy a used car privately, never ever hand over payment until you are absolutely sure about the car’s condition, because you are very unlikely to have any comeback afterwards.