Used car dealer sold me car which had already had cam belt failure and engine damage

[des8]

Why believe what the dealer tells you?
He is hardly likely to roll over and say "sorry guv, here's your money back"

So if you claim under the Consumer Rights Act 2015, he might dispute your claim stating you were not a consumer but a business buying the vehicle to act as a taxi and your claim should be under the Sale of Goods Act 1979
So word your claim under the CRA 2015, but state that in the event this disallowed alternatively your claim falls under SoGA 1979

Of course if he did 12000 miles in 10 weeks the dealer might also start claiming the problem is due to wear & tear

[DE DOGS]

My experience with this problem. Normally when these belts snap it will cause damage to the cylinder head, but not every time, if they are lucky cylinder head off fit a new valve, but if the valve hit the piston that can bend it causing the engine to seize, and on occasion, the engine will need replacing as the valve will drop and damage/score the bore, I believe most manufactures have now reverted to chain drive as there were so many claims for broken belts. In your case, a simple compression test will give the most answers without removing the cylinder head.