I was a Director at a 'Big 4' firm...
...and I would urge you to look elsewhere for a job! From my personal experience, the Big 4, Accenture and McKinsey exist to make money. In order to make money you have to deliver value to the client (only fair), and on many projects this involves obscene working hours, and more time spent away from home than when with the Army. The Big4 also suffer from insane levels of office politics: too many people are spending a chunk of their time working out how to get to partner level, and the competition is extremely fierce. From what I've seen some excellent people make it to partner (and the 350k - 3M annual paycheck) but a lot of scum makes its way to the top as well.
On the positive side, provided you are making your sales numbers then they are typically great places to work in, provided you sign up to the whole "work together, play together" bullshit. I found my time there useful, partly because you do meet incredibly bright people and learn loads of stuff, and also there is a good chance of a client offering you a job (I was offered two roles, both extremely tempting, and without a young family I would have leapt at the chance).
The value of your contacts list is key - as has been stated in an earlier post. Build up your network: attend every piss-up and networking event you can in London (or elsewhere) and meet people. Don't be one of those who throws his business card at everyone in the room, but engage in conversation first, generate interest and then, before moving on (if the person hasn't already asked for your card), arrange to chat later or meet for a beer/coffee/etc. at a later date.
Apart from networking events, what extra-curricular activities do you do? Do you know any headhunters? If so, meet them for a chat. BTW: headhunters/recruitment consultants are only involved in about 30-40% of all jobs being offered, so don't be scared of applying directly to companies. A quick word of warning: the bigger the firm, the more stupid and shite will be their HR department. The quality of HR professionals in the UK is gutwrenchingly low: full-on PC retard.
Joining a Big 4 firm can be soul destroying: KPMG was ok (4 months from their initial approach to me to me sitting down in my office). PWC UK's recruitment process was so shit I walked away (almost exactly a year from first approach (by them), 6 interviews + exams to being told 'we want you to start in 3 months time'). Obviously though, a lot of people love it and thrive.
As for McKinsey and Accenture: I've friends at senior regional partner level, and they can be very good... however I just don't like their business style/flavour (especially the arrogance of Accenture, who in my region are robbing morons of little ability).
I know a headhunter who is an ex-sundodger. He's done a lot of recruitment into the Big4 (mainly in the ME). PM me if you want me to introduce you to him.