Hi and welcome to GMAT Club!
MBA is a pretty powerful degree in the sense that it allows one to transition from an industry such as Education and into Finance, Consulting, or Tech. Combining an MBA and a summer internship you can get the push and the background need to get into one of those industries.... and if you are looking for a job in London, LBS would be the best option since it allows you to do a summer internships rather than many of the 12-month degrees that don't provide that option. You definitely want a summer internship by the way and extra time to recruit.
Usually, recruiters don't care about your undergrad or high school performance. The fact that you got into the MBA program and recruiting from it is enough. There are some benefits if you went to an amazing program such as MIT for example but not much otherwise.
As to Top MBA programs, the best one for Consulting is obviously INSEAD - it places heavily into consulting but it is only 10 months and very competitive. In general consulting is also very competitive so even if MBB's are hiring heavily from INSEAD for example, there is more competition for the best spots, so as you pointed out, certain teams such as implementation or outsourcing arms or SCM are easier to get into.
In terms of preparing for a consulting interview, your best option is to ace interviews, presentations, and cases. They won't be impressed with paper degrees or quick certifications. Your experience may play well - you want to leverage EdTEch but it may also backfire if a bunch of people from Accenture, industries, etc apply for those jobs. You have to be strategic and realistic about Strengths and Weaknesses. You want to demonstrate lots of Senior Leadership/Management/Board exposure and that you can handle the room - that's what often matters for the consulting recruiters.
I am sure
souvik101990 will have much more to share as he was recruiting for Consulting and Tech with EDU background.