You can get your money back provided you can prove the previous owner was aware of the problem and didn't inform you about it before you bought the car. So basically, unless you get them to admit it its very difficult to get your money back..Double check, and do so quickly, beacuse although you don't have as many rights you should still have rights to protect you from this on a private sale.
It depends strongly on if you were misled as to the condition of the car too, but definately worth a call to the local Citizens Advice or something to be sure.
A friend of mine bought a car, but had the pistons go through the bonnet in the first month. Managed to get all her money back off the chap, who was very apologetic.