While there are obviously some differences, the same basic rules apply when buying a used car that also apply when buying any new item. Once you've legally 'accepted' the item you lose the right to a refund; you're only entitled to a repair or replacement. (The trader appears to be relying upon that).
However, since the fault obviously became known fairly soon after purchase, you never legally 'accepted' the car, so you retain the right to a full refund of your money. (The length of the pre-acceptance period isn't defined in law but you must be given sufficient time to check that everything is fully working before you're deemed to have accepted the goods purchased. In a test case, a court ruled that a woman who had purchased a car during the winter, but only found out that the air-conditioning wasn't working at the start of the summer, had never 'accepted' the vehicle as it would have been unreasonable for her to test the air-conditioning prior to that time).
Your local Trading Standards department should be able to advise you further, and may possibly be able to put some pressure on the dealer to offer a refund. Enter your postcode here to find the contact details:
http://www.tradingstandards.gov.uk/advice/