Completely agree with
Mo2men and
JackDaniels.
Let's first understand your priorities. I know your goals, but what comes frist: be in consulting, in London, or you'll take both?
If you think consulting before London, you should maybe consider INSEAD. At INSEAD, there are not only tons of people interested in consulting, but also a lot of sponsored students from MBB. Networking into consulting will be a piece of cake. You could try for London, make it clear as day that you want to be in London, but you will increase the odds of getting into consulting.
If you'd rather be in London than consulting, there's no discussion, go to LBS. 48% of the LBS class of 2014 stayed in the UK, but only 9% where originally from the island. That means a region shift towards UK of 39% of the class. That's a lot. When you look to american MBAs, mostly stay in the US, but they are already there before the MBA.
If you want to maximize the changes of being in consulting in London, numbers say go to LBS. At INSEAD, 41% went into consulting. Out of those, 54% stayed in Europe. Out of that group, 24% stayed in the UK. So your combined chance of being in consulting in London is 5% at INSEAD. For LBS, a third of the graduates went into consulting. As almost half the class stayed in the UK, that means that you'd have a 15% chance of being in consulting in London.
State your priorities, make your choice.