It depends very much what management consultancy company you are applying to. The top level companies, the so called MBB companies, probably only take 20-30 graduates per year, and that figure may well be less in some years (in the London office, and I haven't checked that in the last couple of years). So they can perfectly well be as selective as they like, given the global popularity of the London office. Hence their Oxbridge/LSE focus, plus the overseas prestigious unis and the focus on world class MBAs. You are unlikely to break into these companies with a non-Oxbridge undergrad degree topped off with a Masters, because they get more than enough Oxbridge/LSE + Masters applicants.
However, you can perfectly well get a job with Accenture, Deloitte, E&Y, PWC etc consultancy arms and within 4 years try to move to MBB if that is still where you are aiming.
Then there are a 'middle' group like Oliver Wyman, Roland Berger, Capgemini and smaller, niche consultancies like Newton, 202vision etc which you can try for if you have reasonably strong academics.
Whilst some management consultancies specialise in IT or Finance, and so require relevant degrees, the largest ones deal with all aspects of business and do not require you to have a numerate degree. Bain regularly have specific presentation days for students with Arts degrees (albeit assuming you are at a target (Oxbridge) uni). Degree level numeracy is not necessary in all areas of consultancy.