The car is not of merchantable quality and not fit for the purpose for which you purchased it. That's the Sale of Goods Act, which you should read carefully. If you are not happy with the suggested resolution - and you should wait to hear what that is - then you have every right to 'reject' the car and demand your money back.
I strongly advise you to:
Make a point of being particularly reasonable. Tell them you've had to get a couple of taxis, for example, without immediately demanding that they pay for them.
Put EVERYTHING in writing, preferably e-mail. If you have a phone conversation, e-mail them straight away with a summary of that conversation "in the interests of clarity".
Ask for time scales (in writing), and chase them up regularly (in writing).
Your contract is with the supplying dealer and the onus to repair your car is on him. In involving the warranty company, the dealer is just trying to minimise his losses.
If there's no satisfactory resolution offered, then you will need to register a claim in the small claims court. The limit is £10,000 and the process is straightforward. More here:
https://www.citizensadvice.org.uk/law-and-rights/legal-system/taking-legal-action/small-claims/Good luck, and please keep us updated.