Hello!
I just want to say I fully sympathize with you missing out on all these opportunities. There are a lot of widening access opportunities but the support in Scotland from all schools for Oxbridge is dismal so a lot of students don't even hear about these opportunities until too late when they're the ones who would have benefitted from them.
I actually received an offer for that exact same course in Edinburgh last year, so first of all congratulations! You are absolutely right, Edinburgh is an excellent university. Not only that, it may not seem very real now but graduating with minimal debt is a big deal. I did take the hit and accept my Cambridge offer, and a lot of people will of course, but I would consider this deeply as it is a big financial decision (doesn't help we have to make it so early in life but that's the way the cookie crumbles). Regarding net working and things, it's true Oxbridge is useful for that but these days with the internet you have access to more opportunities than you realize. A university degree is what you make of it, so a driven and proactive student at Edinburgh will get farther than a passive and complacent student at Oxbridge. Gone are the days when simply the Oxbridge name opened every door, you need to work hard to get the most out of your degree and I'm sure you will do that no matter where you go.
In terms of what you should do - I know you said taking a gap year is not possible but is there any way you could take a year out to work? As a Scottish student you are a year younger than all your English peers at this point anyway. If it's not feasible then no worries, but I do think this would give your Oxbridge application the best chance. As others have said, it is very rare for them to consider applications once someone is already at university. I would e-mail them and ask exactly what they mean by exceptional circumstances but how they see it is you already have a university and a course and it would be bad form for them to encourage student to sacrifice that for Oxbridge.
Regarding risking your Edinburgh offer - I don't know when your birthday is regarding age cut offs, but when I received my offer they actually suggesting the could defer it since I was only 16 in the actual offer letter! I would e-mail them and ask if you can defer the offer, and from my understanding if you have an offer for an other uni you can still apply (I got into Cambridge when my offer from Edinburgh was already firmed).
If this is unsuccessful, I would actually say you could just reapply to the course again on a gap year. Depending on your grade profile you are very likely to get in. Because of fee status, Scottish universities are much less competitive for us than English students. When I went there on open day the lady practically told me that anyone with 5 As at Higher is guaranteed an offer. And sure enough a week or so after I send my UCAS in October I got an unconditional.
I hope that all makes sense. Let me know if you have any questions and I hope it works out for you.