You're not going to be disadvantaged based on your background. Sure they look at your GCSEs within the context of the school you attained them at but it's literally impossible to improve on what you got. You don't need to be a prodigy, you just need to be good.
All you can do now is write a decent personal statement (doesn't have to be amazing, but the better the better obviously) and prepare for the PAT. People usually don't finish the whole thing in time so one thing to work on is being able to answer as many questions as possible. The better your base knowledge, the more you do it, the quicker you'll become. Skip questions if you're taking too long.