Plenty of people at Exeter are not from private school, so don't worry!
Just remember it cuts both ways - I didn't attend a private school, but am still in a fairly fortunate position financially. Perfectly innocuous comments that I made would be seized upon and I would be made to feel bad just because of an accident of birth (how is it my fault that my parents have worked hard and were able to see me through uni?!). An example would be how one of my second-year housemates was complaining about how she didn't like the taste of the tap water, and so I suggested that she bought a water filter (doesn't have to be anything expensive like a Brita - they all do the same job) as it would make the water taste a bit better. She knew I had one and made a really catty comment along the lines of "Oh, well, we can't all afford that".
So certainly nobody should be nasty to you because of any lack of wealth, but equally, make sure that you don't set upon people for the opposite reason (not that I'm suggesting you would - just saying).