David - thanks for the positive feedback on the site! Re. getting into McKinsey, BCG, Bain... the issue is really that more people aspire to join these firms than any other firm, so they have the pick of the crop so to speak. They also recruit in relatively modest numbers each year, so it tends to be the "best of the best" that get employed and both your academics and your "fit" with their culture can play for or against you.
I haven't been involved in graduate recruitment for a while, but you could assume a strategy firm like this might interview 100 graduates each Milkround in their London office, with a view to hiring 15 people. So first question - do you think on paper you look like one of the top 100 graduates they are going to get applications from. They will be inundated with candidates that have a First or 2:1 from Oxbridge, that have done internships in consulting or investment banking. So even being amongst the 100 graduates they choose to interview is a tall order. And then obviously your case study skills have to get you over the next hurdle to make it to the final 10 or 15 that are offered.
I would suggest contacting the careers service at both UCL and King's and finding out from them how many of their graduates have joined any of these firms in the last 5 years. This will give you a real indicator of your chances with such firms.
For those that don't make it into these firms at graduation, my advice is always to try and do a strategic role elsewhere (combine Accenture, E&Y, PA Consulting, PwC, etc. and there are actually several hundred strategy consulting graduate roles to fill each year in the UK). Get one of these on your CV, perhaps do an MBA from a top business school and then a transition to the top strategy consultancies is not out of the question later in your career.
Hope this helps & good luck. Tony