Welcome to the world of consulting, the land of interesting work, long hours and occasionally meeting people with entries in the Guinness Book Of Records for having the worst social skills in the world. On the bright side, they don't last long.
To actually attempt to answer your question, you're not too late to become a consultant and the MBA is probably the minimum entry requirement given your non commercial background.
What I would suggest you do now is research the companies with a particularly strong public sector focus and steer clear of "systems houses"; you've never built an IT system so you'll have to market yourself slightly differently.
As an initial list I'd have a think about Hedra, PA, Atos, CGEY and the like. There are regular recruiting events advertised on Top Consultant, you need to go to one of these and actually talk to people - don't rely on online adverts and don't rely on posting your CV because frankly recruiting departments get thousands a year and if there's an obvious reason not to talk to someone (e.g. not having consulting experience) it'll go in the bin. Find a couple of agencies, no more than that, and physically go and see them too - again, in your position you need to be able to get face time with people and not just send CVs around. I'd suggest Prism, Huntswood and outfits like that.
Think carefully before you go to the recruiting fairs about what your own personal elevator pitch is - if I was recruiting, I'd be asking myself what an ex-teacher with an MBA can bring. There's a very good answer (comms skills, analytics, MBA etc etc) but you need to work it out for yourself. You also need to have a think about what package you're looking for - I recently recruited someone for 30k who'd previously been on 20k at Qinetiq because she had no obvious experience but was clearly very good and motivated; she's doing a great job, is worth every penny and I have every confidence will progress up the ranks at a sensible rate. Keeping it virtual does not play to your strengths - good consulting is heavily dependent on having good communication skills (see earlier in this thread for a particularly poor example of why!!!) and as a 29 year old teacher you should be able to bring those along with a bit of maturity.
Don't expect this to be easy and don't get discouraged when you inevitably get knockbacks - fact is, you're not going to get into McKinsey but then neither could most of the 100,000 people at Accenture and they're all doing very well for themselves.
Good luck, and feel free to ask any other questions. We're not all complete clowns around here.