I am going to make an assumption that your A-levels are not mostly A grades.
If so, that would be the only obstacle I see as many of the Big Law firms have a AAA or AAB requirement.
But that’s not to say you couldn’t get open day with them or paralegal with them. That restriction typically only applies to vacation schemes and training contracts. You may be able to get your foot in the door and let them see your potential beyond 2 years worth of grades when you were much younger.
Even if you did struggle with Big Law, then you could always train with another international firm and then move as a qualified lawyer.
Given your connection to Cambridge and later at Oxford, I would recommend utilise their exceptionally strong alumni networks at both a central university and college level.