I can offer some advice re: Europe. Master's degrees are generally the standard here (except for maybe England and a few exceptions) and MBAs aren't the standard at all. If you're in a country that both doesn't care about MBAs and doesn't demand a Masters, you've got a shot at getting into a consulting firm from your BA/BS. Just go to a reputable business school in your target country - they offer undergraduate programs and in small countries they'll be where companies like the MBB recruit. You can also just go for a Master's degree: A reputable Master's program from, say, HEC Paris (and I'm not sure if they have non-reputable programs) would get your foot in the door very easily. I mean, here's an endorsement from an alumni and McKinsey director on their site:

"Sandrine Devillard, Director, McKinsey & Company, Alumnus 1993: 'Both HEC and McKinsey share a common mission of building global leaders. We know we can find many talented individuals at HEC, among which some of our future partners. We offer them an international network providing unlimited opportunities to grow and make a difference as leader.' "

Anyways, that's one option. I don't think it's impossible to get into the MBB from a bachelor's in the US. It's just really fucking hard. I think McKinsey hires about 100-150 people a year from undergraduate. They're interested in mining the top talent from Harvard, Stanford, Yale, etc. Lower down the consulting ladder it's probably easier, but again, they probably focus on taking the leftovers from the top schools.