Hi, living in Germany I'm not really qualified to state concrete facts as applies to either UK law or the Mazda warranty. However, to indicate that your post isn't being ignored I'll offer you my own personal opinion and leave it to others to contradict anything that I might say or, alternatively, offer their own and perhaps more qualified opinion.
Looking at the Mazda UK website under the tab 'Warranty'
https://www.mazda.co.uk/buying-owning/buying/warranty/ it quite clearly states that they offer a 3-year
manufacturers warranty. Please note - manufacturers warranty - and not dealer. This warranty can be extended with three levels being available but in my opinion this warranty is offered using independent and specialist insurance companies with items covered under the extended warranty being very much less than the initial manufacturers warranty. Either way, it would seem from your posting that this extended option was not chosen by the previous owner and, therefore, even if it was transferable to subsequent owners (as is the initial manufacturers warranty) neither the initial manufacturers warranty, and thus a claim against the manufacturer, or the extended warranty, and thus a claim against an insurance company are options in your case.
In my opinion, any reputable dealership - and it may even be covered by legislation in the UK under consumer laws - should/has to provide a limited time guarantee on any vehicle sold. If this is the case, and I would imagine that it is, then any claim that your submit has to be made against the seller. If they seek advise or an opinion from the manufacturer then that of course is up to them but in practical terms has got absolutely nothing to do with you. For example, the dealer may say that repair costs amount to X pounds but due to the age of the vehicle and it's full service history, the necessary repairs should not be required and that the manufacturer should really be paying a portion of it. As I said though, any dealings between the seller and the manufacturer outside of the vehicles manufacturers warranty period have got nothing to do with you.
You do not say exactly when the vehicle was purchased as of course if this was more than 6 months past and the dealer only offered a 6 months used vehicle warranty then your case is very much weakened I would say. Either way, with the information that you have provided and with the likely costs involved I would immediately contact a solicitor if only with the view to seeking advice and if it seems that your case against the seller is reasonable then ask the solicitors to handle the case on your behalf. Not only will the know the correct way to proceed applying the due amount of pressure on the seller, but also they will be able to claim costs - including extra costs incurred by you - for the work done.