A Brit saying dude would actually be funny. Your grammar is better than most, trust me. Also, I doubt that most people know how familiar you are with British politics and the educational system; it's not as if you go around talking about it all the time. Who tells you that you have an accent anyway?
Random note: you know what I'm thinking of right now? That joke about the dumb American in the resturaunt "in the groove man, in the groove"