You can indeed work up to 20 hours (at least now, though they change immigration rules all the time). It can be hard to get a really good job however, even in London, since a lot of places don't want to deal with either student workers or applicants with visas. You should check out temp agencies, an office job would probably pay 8-10 pounds/hr. A retail or restaurant job would be 6-8 pounds/hr. I don't think it's too hard to get a restaurant job in London (pubs are more fun to work at but you don't get any tips). I actually found it really helpful waitressing during my masters because while the base pay was only about 6 quid/hour I also got tips, free food, very flexible shifts, etc.
If you want to try to get any kind of editorial work here (which I do now, and which pays much better!) you'll have to show you can deal with British English spelling, punctuation, etc. (it is really easy to pick up, you can use it for your academic papers so you have written examples to show people)
I really wouldn't worry so much about showing that you're "unique". I think that's more a concern for American schools, finding unique and well-rounded individuals, etc. Here, I think it's more about -- can this person do the work? are they really interested in this subject? what would they do with this degree? what would they add to the program?
As a TA I constantly tell students the first rule of an essay is: answer the question. I know it seems obvious, but if you find yourself struggling to come up with something brilliant, go back and really look at the question posed to you. If it's something like, why do you want to join this program and what do you hope to get out of it, then they probably don't want 500 words on your personal trajectory since high school -- you can sum up very briefly your ability to overcome personal challenges and/or mistakes and go on to a more forward-looking statement.
Forgive me if I'm overstepping but -- while I do have a great deal of admiration for the way you have worked so hard and come so far, I don't think it's necessarily helpful to think this is the only "unique" thing you can present about yourself. Your posts show a great deal of enthusiasm and curiosity about the wider world, a desire to fully immerse yourself in a foreign culture, a great interest in your academic field, etc. I think these are the kind of positive qualities that would be great to highlight -- and then you can say that you are confident in your motivation and ability to do the work given how hard you have worked previously to overcome the challenges in your life.
(I say this as someone who came from a rather poor/dysfunctional background myself, I think even when it's a really important part of your story you still need to make it relevant to people. What have you learned from that experience that you would apply to future studies? Does it have any particular relevance for what you're studying? Basically, why do you think it's important that people know this about you? Obviously it shows great strength of character, but you're still applying for an academic post, and I think for UK postgrad they care a lot more about your capabilities than your overall character.)
Anyway, I say, why not apply to Oxford? I told myself there was no way I would get in, and then when I got here and met students both at my uni and at Oxford I thought, geez, I totally should have applied...
