If you search the archives of this board you will find several previous threads from people in the same situation as yourself.
Did you apply to Oxbridge this year and get turned down? Or did you not apply? If you did apply, did you get any feedback from your interview? (Your school may have some, even if you don't.) Have a look at this and see if it explains areas where you can improve.
Don't be too down about getting rejected from Bristol. Almost everyone gets at least one rejection. I've never heard of a university having 'rounds' of applications as you describe. I always thought it was just apply through UCAS, then the uni either accepts you, rejects you or asks you for an interview.
As everyone else has said, re-applying is taking a big risk. There is no guarentee you would get into York, or any of your other choices, again. Are you prepared to take this risk? One of my best friends got in through re-applying, but I am sure many people will post about people they know who didn't get in.
Apply for the course and not just the place. If you go to the interview just because you want to be at Oxbridge, and not because you want to do the course, you will be rejected. They like people who are very enthusiastic about their courses.