This can vary by firm, and by office. In general, for London, the policy is pretty equal at MBB. To answer your questions:
1) MBA sponsorship is more or less guaranteed as long as you don't have a history of underperformance. This means that as long as you are meeting expectations and are "on track" for normal promotion schedule, the firm will sponsor your MBA.
2) In general in the US, an MBA is more or less an expectation. In London, on the other hand, it's definitely not expected, nor would I even say it is especially encouraged - the option is made clear and people are supported if they choose to apply, but there's no expectation at all.
3) It won't affect it much - the only difference is if you leave before you get promoted to the post-MBA position (Associate at McK, Consultant at Bain/BCG), you will return as a post-MBA hire. However, it won't particularly accelerate your career - if anything, it will slow it down as the 2 years you spend in your MBA you could have been working and would be close to the Manager promotion. However, the MBA can offer many other things - mainly soft skills, some business knowledge, and network - which is why many people still choose to do it.
Hope this helps!