What people often don't realise is there's a massive upwards tail of private schools - from those that are only a little different from a good comprehensive / grammar to public schools like Eton that are as distant to us as they are to you. I'm not sure which you are going to, but the following is based on my own experience (state primary school, years 7-13 at a normal private girls' school on a scholarship).
I suspect that unless you're incredibly unlucky you'll have very few problems being accepted. Private school kids (as a generalisation) are pretty secure with their own wealth, slightly naïve about social interaction (they don't need to grow up so fast - this can also manifest itself in other, less positive, childish ways, but on the whole no more than any school), and a bit curious about the outside world, so if you're friendly to them they'll be friendly to you.
The main problems you may find will probably be unintentional. Namely, people having more money to spend on social activities, and people being naïve about other peoples' financial situations. If the first happens, just be honest with them - if they're decent people they'll understand, but if you don't go out and don't explain why they may think you're being antisocial. With the second, most are pretty educated, but not having exposure to anyone poorer than them means it's not uncommon for a minority of private school students to come out with comments like 'I didn't know council estates existed in real life'. Just remember they probably don't mean it badly - they just don't know.
If you do meet any vindictive snobbery, shrug it off. If it's bullying, tell a teacher, but if it's an offhand comment, either ignore it or respond as you normally would. Whatever you do, don't assume snobbery that isn't there, or respond with inverse snobbery, as if other people overhear it they might be put off befriending you. I only say this because people from private schools get quite a lot of inverse snobbery and whether it's justified or not tend to have no time for it.
With that slightly odd point out of the way, you should be fine. I won't lie, you may well meet some absolute ****s, but they aren't that common, and state schools have equivalently nasty people anyway. Have fun, enjoy the academic atmosphere, and find some decent friends - the truth is that most of them probably don't care a bit about your background, except for curiosity, and if it doesn't matter to you it won't matter to them.