You will need to spend time looking at lots pf private ads for the well looked after, low mileage motor - not autotrader, but local ads in supermarkets, free papers, post office, or ask friends, colleagues etc.
Chances are any dealer selling a car at about £2,000 will actually be selling a car which cost a few hundred quid at auction - unknown history, no paperwork - basically unless you know what you are looking for it could be a massive gamble.
If you don't want the expense of a cambelt change you'll need a car with chain drive. Unless I knew a £2,000 car had had the cambelt changed before I took ownership it would be the first job I had done.
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