What was the mileage on the car when you bought it?
I would be inclined to ask the warranty company to pay on the basis that:
1) At the time you bought the vehicle the manufacturer's interval for belt change was 36k miles
2) The mileage at the time of purchase was XXk miles (fill in the blanks, as per the question above)
3) The selling garage has an obligation under the Sale of Goods Act 1979 to ensure the car was fit for purpose
You can argue that they - as the seller - should have ensured the necessary work had been carried out and clearly hadn't done so.
There is a limit on the time period, which is 6 months, during which they have to rectify the faults. Outside that time they are not obliged to act, so you may find they will refuse or may make a 'part payment' offer.
They may also suggest that you should have checked the paperwork when you bought it, to assure yourself all necessary work had been done. However, you appear to have relied on their information in good faith, so that isn't a strong defence.
Be prepared for a bit of a fight, but good luck and keep going!