Not attending a target undergrad school

Hi all, I am currently getting my BSc in Business at a good London university although not a target one (Birkbeck). I wasnt able to get into a top school due to my poor A level past but nonetheless from now on I am willing to push myself to the maximum so I can get into IB. What would you all recommend me doing to increase my chances once I graduate? I thought about doing a MSc in Computer Science at Imperial afterwards or in another subject Finance related at a top London university. I usually consider only institutions in London as I got used to live here and wouldnt really imagine myself living elsewhere in the UK but I dont see why I wouldnt attend a top university like Cambridge or Oxford for my masters regardless of that. I want to do a MBA one day at a top school but I would prefer doing it in the US but I plan on doing that well later once I acquire valuable working experience. I can imagine I am at a disadvantage as English is not my first language but I am trying my best to erase that disadvantage and be able to compete against anyone.