That's very realistic if you join uni at around 18-19.
That's impossible because although MBA takes 1 full year to complete (if you do a FULL MBA that is), you need to have at least 3 years (Cambridge MBA says their MBA students have an average of 5 years post grad work experience) of
post-graduation work experience involving management opportunities. Now, just to mention, that part-time or full-time jobs at bars, cafes or any similar stores (eg. cash tills at Sainsbury's) usually fall out of this category. Though I'm not sure if an administrative
clerical job would count.
Realistically, you need to gain real-time work experience following your graduation for 3 or more years, showing an active interest in management or related disciplines (favourably involving in team management) and forging good contacts with your seniors and other co-workers. This will ensure you will have good peer and supervisor references while applying for a MBA in good unis. Maybe you wouldn't need your company to sponsor you, but in a lot of cases they do and you can put that £15k-£30k in your account instead of paying it to the uni concerned.
As for the chartership, it's better to work for a company which is approved by the concerned engineering body, example IMechE. This way, all your work will be monitored and after a minimum of 2 years, you can apply for chartership. However, in most cases, chartered engineers have 3 or more years of monitored work and training following their graduation before applying for chartership.
I have some case studies of chartered engineers, and you can also get them if you navigate websites like
www.imeche.org or alternatively, PM me and I will send them to you.
Your course should be a MEng to satisfy the academic requirements of chartership, or you need to have a post-grad qualification to fulfill it. Maybe you know this already.
So it is likely that you graduate at 22,
become a Chartered Engineer at 26-27,
do a MBA at minimum 25 (but usually around 26-28),
and start your own company in your late 20s or early 30s (at the earliest).
But just because you have MBA and a CEng status doesnt mean you can start your business straightaway, so maybe you may feel you need to have more experience of how a company functions and how you can manage it before starting your own. In that case, you may as well start your company in your late 30s or even 40s!
Wish you all the best!