There is absolutely no requirement to have attended public school before joining the RAF.
In fact most public school careers masters will try to persuade you to follow another career path - they view the RAF as being frightfully common and plagued with loutish grammar school oiks. Or, even worse, those who have been to a 'comprehensive school', whatever that might be.
If you have been fortunate enough to have been educated at a public school, you will find that during basic training you will be viewed with jealousy disguised as mockery by those oiks who speak with pronounced regional dialects - and who " 'av 'us dinner before 'us tea".
OK - not serious. The RAF is more interested in what you have to offer than what your background happens to be. Don't sweat on it!