What Should I Do?

I am a sophomore at a top public high school in Istanbul, Turkey (I have Turkish citizenship). My main objective is to break into IB in London. However, I am at a crossroads because I am not sure which University is best for me. It is mainly between Bocconi and the UK top 5's, but this is where it gets complicated. Of course, my top choice is to attend either LSE, Oxford or Cambridge but the tuition looks out of reach for my family, maybe just scraping through. On the other hand, Bocconi is half the price and more accessible (easier admission). So, what are the possible ways I can break into IB in London? Will the Italians be ahead of the internationals at Bocconi?