http://www.which.co.uk/consumer-rights/advice/the-second-hand–car-i-bought-has-a-problem-what-are-my-rights
If there’s a problem with a second-hand car soon after you’ve bought it – for example, the car develops a problem you wouldn’t expect for the car’s age and mileage, or it turns out not to be what you’d been led to expect – you may have the right to reject it and get your money back.
If you bought the car after 01 October 2015, you only have 30 days to reject a second-hand car and get a full refund under the Consumer Rights Act.
During the first six months after purchase it’s the responsibility of the seller to prove the fault wasn’t there, not for you to prove that it was.