I'm not sure I 'buy into' the idea of target / non-target schools in the UK to be honest. Coming from Oxbridge or LSE is great, but counts for little on its own. Having a perfect CV with plenty of experiences, courses, knowledge, enthusiasm etc. and researching the company, along with going to a good uni (like Bristol) is enough. Half of it is luck of the draw, to be honest, which is why applying to as many spring weeks as possible is a good idea.
Edit: I should say, I'm currently interning in a directly adjacent industry to IB, and far far fewer other interns are from Oxbridge than you might think. (And lots of people from Bristol!)