As you bought the car in June 2015, the relevant pieces of legislation would still be the Sale of Goods Act 1979 and the Sale and Supply of Goods to Consumers Regulations 2002. These have now been replaced but the law applicable at the time of sale will be the one that you can rely on.
The law states that the goods must be of satisfactory quality, as described and fit for purpose. If they are not, you will have certain legal remedies against the seller. There is no protection against fair wear and tear, misuse or accidental damage, faults that were pointed out at the time of sale or if you change your mind and no longer want the goods.
If the goods are not as described, of satisfactory quality or fit for purpose, you have the following rights:
1. Reject the goods and request a refund - this is only possible if the rejection occurs within a 'reasonable time'. This period depends on the circumstances, although it is generally accepted to be within the first month after purchase, so must not be delayed.
2. Ask for a repair or replacement – if you are too late to reject the goods, you can ask the seller for a repair or replacement without causing any significant inconvenience.
So legally you are likely to be too late to ask for a refund. They can choose to offer that option but they are unlikely to be bound by it. In terms of the offer given to you, whether you can rely on it will depend on whether a legally binding contract was formed. In the circumstances that is unfortunately unlikely. They made you an offer of £48k, you rejected that and counter-proposed £50k. This offer was never accepted by them so there would be no legally binding contract in place. In effect, there is currently an offer for £48k by you, which is open for acceptance by them but unless they go ahead and accept it, there would be no obligation on them to offer a refund for the £48k as that offer has now been rejected.
Therefore, you are either going to have to continue negotiating until you find a price you are both happy with, or proceed with the other remedies, such as getting a repair or replacement.
This is your basic legal position. I have more detailed advice for you in terms of the options you have should they refuse to consider a repair or replacement too, which I wish to discuss so please take a second to leave a positive rating for the service so far (by selecting 3, 4 or 5 stars) and I can continue with that and answer any further questions you may have. Don’t worry, there is no extra cost and leaving a rating will not close the question and we can continue this discussion. Thank you