The law is 100% on your side . . .
The dealer failed to discharge his implied duities under section S14(2) & S14(3) of the Sale of Goods Act; ensure the vehicle is of satisfactory quality and fit for the purpose it was designed for.
Also, he 'may' have breached S75 of the 1988 Road Transport Act by selling a vehicle in un-roadworthy condition. This is a criminal offence on his part!
So, the process to the letter of the law is either :-
1)Totally reject the vehicle and demand a refund
2)Allow a repair (this does not stop you asking for a refund later)
The dealer is entitled to repair the vehicle in a 'resonable' period of time, if it is uneconmic for him to do so - full refund!
Some dealers can be slippery types . . . but rest assured you are looking at a full repair (not even related to the warranty - this was not a warranty failure it was a breach of the condition of sale!) or a full refund.
Furthermore, you are looking at full expenses whilst it is being repaired. So keep receipts for everthing car / travel related. Chapter 54, Part IV, Section 29.6 of the SOGA states :-
"Unless agreed otherwise, expenses of and incidental to putting the goods into a deliverable state will be borne by the seller"
Good Luck! Although you don't really need luck in this case