This is what "Which" has to say about it:
The second-hand car I bought has a problem, what are my rights?If your second-hand car develops a problem, you’re not alone. Regardless of whether the banger you bought was from a used-car dealer or a friend of a friend, you do have rights.
Rejecting a second-hand carIf 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 its 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 any time after 1 October 2015, you have only 30 days to reject it and get a full refund under the Consumer Rights Act.
Cars bought before 1 October 2015 would have come under the Sale of Goods Act, which has now been replaced by the Consumer Rights Act. This stated that you must have rejected the car within a reasonable time frame (probably within three to four weeks – less if it was an obvious problem).
If you reject a second-hand car bought in the UK, you must stop using it immediately.
Returning a second-hand carIf you're past the first 30 days but a problem has arisen that you think would have been there at the time of purchase, you're entitled to ask for a repair or replacement free of charge.
In most cases this will be a repair, as whoever sold the car to you will usually be able to prove that the cost of replacing it would be disproportionate.
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.
But after the first six months, the onus will be on you to prove that the fault was present from the day you bought the car.
If the attempt at a repair or replacement is unsuccessful, you're entitled to a refund.
But the car dealer can make a deduction from the refund after the first 30 days for 'fair use'.
https://www.which.co.uk/consumer-rights/advice/the-second-hand-car-i-bought-has-a-problem-what-are-my-rights