Check this post out on Reddit.
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A few thoughts from my rejection experience;
1. If you paid by cash and you have already sent them a strongly worded, accurate letter/email stating the various elements of the consumer rights act and itemising the time and dates of errors with the car and the longevity of the problems, then a solicitors may help, but bear in mind each time will be £300-£500. The dealers are aware most solicitors letters are just 'hot air' because they already know you're in your rights to reject the problematic vehicle, but they are 'protected' by the limits of the small claims court. If you paid ANY amount on credit card, even just a holding deposit, you are protected up to £30k purchase, sadly, these cars will no doubt be well outside that, but worth knowing.
2. If you have bought on finance you are in a much stronger position, as unlike the actual car dealer or manufacturer who have little to no governance, the finance companies are regulated by the Financial Conduct Authority, so raise a complaint with the finance company (they will ask for all correspondence with the dealer, manufacturer and all of your evidence) and tell them you will be raising this with the FCA and Financial Ombudsman if a solution is not found within fourteen days. Naturally if you can afford a solicitors letter, as above, then its a often a quicker move but again no absolute guarantees, but it may take out some of the leg work.
3. Rejectmycar.com offer a good service but charge you 10% of the cars value. My refund ended up being the same as it would have been with rejectmycar, but it took me two years of stress and anguish, they may have got results earlier, or not, but they would have obtaining a full refund rather than the less offer I accepted, so it worked out the same, plus the stress and driving a faulty car for two more years.
4. DO NOT under any circumstances allow the vehicle to go back in to the dealer workshop. If they apply any updates, they will say 'cured' (even if not) and this puts you on the back foot especially with the Motor Ombudsman. If there are ANY official recalls sent out, then do not have them performed, as an official recall is a clear 'win' in that the car was sold in unsatisfactory condition - this is why manufacturers tend not to send them out unless they really have to.
5. The mistakes I made - not itemising everything correctly and sending to the dealer (TW Whites and Sons), making comments on forums when the car behaved itself like (oh today the car drove without fault - the dealer found out whom I was by seeing photos I posted on my car on a forum and facebook group under my username and cut and pasted this and used it in their case with the TMO), allowing the car to go back in for repairs multiple times (one of which they sneakily uploaded a software update which later became a recall, if I hadn't let it go in, I would have won my TMO case instantly a year ago), accepting a 'goodwill gesture' - they later used this claiming because I accepted the goodwill I gave up my rights to return the vehicle (thankfully the TMO did tell them that was BS but still was another thing I had to argue), and record any calls and meetings you have with any dealer , service manager or customer service agent - these were absolute invaluable to me and won me my case even know they were made under 'stealth' conditions. If a dealer makes a comment like 'we have another one like yours with the same issues' and you get to record it - you will win your TMO case because it shows there an inherent problems with the cars.
Hope this helps