It's infuriatingly ridiculous that the dealer wants to hit you with electronic diagnosis when the physical evidence of a broken N47 chain is there for anyone to see. As long as the car has been serviced on schedule (or to tlc), refuse to pay for any diagnosis and demand the supplying dealer replaces the timing chain and all broken components FoC holding the threat of Small Claims action against him if he doesn't.

Your claim will be that the engine of the car and its servicing were never "of satisfactory quality" for that to have happened in 3 years and 73k miles. Law here: www.honestjohn.co.uk/faq/consumer-rights/ (If it was under tlc, the reason it will have failed is the inadequate number of oil changes under tlc.)