Hello,
Thank you for your in-depth post and insight into your profile and life and what made you who you are today... all relevant and b-school applications can be quite a process of self-reflection so you've started that part off on the right foot.
It sounds like OVERALL you have a lot of ambition, drive and determination... and I applaud that as well as your aim to plan well ahead for the coming years. In terms of profile feedback, since you're considering b-school a few years out, it is all a bit tougher to analyze at this point since the GMAT will be relevant as well as how you finish your current degree, and then what you do afterward will make a large impact in your profile assessment at that point and your admissions status.
To answer some of your q's as best as I can...
1) How important is the undergrad institution you attend? I checked the HBS website and the institutions they admit students from and whilst the are a number of less well known U.S. institutions which aren't Ivy league schools, i'm not sure if this is still the case with UK institutions? How are UK undergrad institutions which are not Oxbridge/ Imperial/ LSE/ Warwick viewed? Specifically the likes of Nottingham, Southamton, Manchester for anyone who knows.

-It is relevant and of course a plus the more competitive the institution is, but that said a) the schools are assessed on their own merit and within a given country, b) your performance at the school and classes taken etc is also very important, and c) you can't really change this at this point so I wouldn't stress too much over it.
2) Will attending a "feeder school" MSc offset my undergrad school name? Or is the weight of undergrad school extremely high? What about a startup can that offset the undergrad institution?
-Not 100% sure I'm following the entire question but I was a little surprised to hear your aims of applying for a Masters in Finance or similar, and then the MBA. Many people go for one vs the other. Sure occasionally people have a similar masters and realize they need the MBA to transition into another area or more into general management, but if you are starting off now trying to decide which way to go, I'd do some more research into Masters vs MBA and jobs opportunities, curriculum, etc and likely focus on one, just my 2 cents.
3) How is the Accounting and Finance double major undergrad viewed by B-school admission officers? Is it viewed favourably, unfavourably compared to the likes of Econ, Engineering, or are they neutral on the whole?
-Overall I'd say neutral though it will show hopefully that you can handle the rigor of b-school which is a plus.
4) Will getting an MSc negatively affect my chances of getting an admit for an MBA? If so I might have to reconsider.
See my points in #2.... it's just not often that people plan for BOTH.... probably some additional career research will be useful in the coming years. You have the time to really explore and identify where you see your path leading and then decide which degree/option will best get you there!
5) Mature undergrad students, how are they viewed in general? And what about if it's someone with a background story like mine?
5) What are my chances based on my "plan" (lol)? I'm thinking if I get into Imperial/ LSE/ Oxford, I can apply to the Harvard 2+2, seems a solid plan?
Really depends upon a variety of factors and see notes above.
6) Any other tips about how I should actually approach this within the next 5-6 years, perhaps I am going about this in the wrong way? Any help appreciated.
Think we've covered it above... additional things to keep in mind are that schools will certainly value leadership; teamwork; people & project management skills; international experience; analytical thinking; making an impact; community involvement, to name a few!
Wishing you the best, and let us know how else we can assist.