Faulty Car Bought from a Private Seller

Private sales: you have fewer rights than when you buy from a trader. The car must be as described and the seller must own it (have good title), but it does not have to be of satisfactory quality or fit for purpose unless the seller said so. Some traders pose as private sellers to avoid their legal obligations; report this to Citizens Advice. source

If the car turns out to have outstanding finance, a private buyer who bought in good faith, without knowing about it, may be able to keep it (conditions apply, e.g. being the first private buyer); otherwise seek a refund from the seller (CRA 2015 if a trader, Sale of Goods Act 1979 if private). source

Dealer sales: goods must be of satisfactory quality, fit for purpose and as described (Consumer Rights Act 2015). If a car fails these tests you have a short-term right to reject and get a full refund within 30 days, counted from when ownership, delivery and any installation are complete (s.22). source

For the first 6 months a fault is presumed to have been there at delivery unless the dealer proves otherwise; after 6 months you must show the fault was already present (s.19(15)). source