I don't know if you appreciate how Warranty works. Under the law the seller is responsible for the condition and function of goods or services they sell, and that is enshrined in the Consumer Rights Act 2015. These statutory rights are automatically applied when you make a retail purchase, and only apply between the purchaser and the seller.
Some provisions of the CRA also apply to private sales, but generally product breakdowns or wear an tear do not. If a private seller miss sells an item, for example claiming a caravan has no damp problem, but when the buyer gets the caravan its riddled with damp issues, then a common breach of contract has occurred and you could pursue the seller through the small claims court for values under £10K
However many product manufacturers also offer an additional manufacturers warranty, which is entirely separate to the CRA and can never supersede the CRA, You will need to read the terms and conditions of the manufacturer's warranty to see what is covered and how to access it. These warranties can often be transferred to a new owner by contacting the manufacturer.
In theory if you have a valid claim under the manufacturers warranty, any dealer signed up to the scheme can undertake the work, but they are not legally obliged to. And sadly many try to decline such work not becasue of the parts ( which should be supplied by the manufacturer foc, but becasue the labour rates the manufacturer pays is well below the dealers labour costs so dealers will lose money doing manufacturers warranty work.