Used car crapped out on us after 40 days - where do we stand?

We bought a car recently from a small used car dealer. It was old and cheap (£1400) so did not come with a warranty. After less than two months (but, crucially, more than 30 days) the water pump broke so it overheated. It now needs a repair job equivalent to roughly the price of the car. We contacted the seller who said he wasn’t liable due to post 30 days but we have found stuff online that disagrees. We can’t afford the outlay to fix it so it would otherwise be scrapped.

We offered the guy the chance to give us a partial refund or try to fix it which he refused. How do we go about enforcing our rights? Supposedly he’d need to prove the fault wasn’t there when we bought the car.