if you come from another MBB firm, you might be considered an experienced hired already after 2 years of working experience. If, as in your case, you come from a completely different industry, you might need slightly more, such as 3-4 years experience. This what happens in average, however, note that there might be differences or exceptions among the different offices.
If you pass the selection as experienced hired, you might start with a "junior" title. For example, at McKinsey you might start as "Junior Associate" which is a non-official position that will last only for a "trial period" of 6 months or similar, before you get officially to the next step, in this case, to "Associate" (that in BCG is call "Consultant").
As a matter of fact, less people join the firm as experienced hire. However, talking with a few colleagues from the HR department, I discovered that there are just fewer applications of this kind and the people come less prepared to the interview (maybe thinking that their experience is enough to guarantee them a spot).
If they like your CV, they like your CV, full stop. They will then suggest you in which position they see you better. Will an MBA increase your chances to get in as an experienced hire? Certainly yes! However, you can also try to apply now and in case you don´t succeed, you apply again after the MBA: 1-2 years will have passed and your CV will look brand new.
I hope I´ve been helpful and made a bit of light on this point.