Greetings from Warsaw!
I'm in Poland visiting some relatives on vacation after graduating from U of I. A friend just asked me for some advice on applying for a Gates Cambridge scholarship so I figured I'd post this here too. I didn't have any luck with the application, but I can still give some advice that might help you decide if you want to apply for one, or if you do apply for one, things to consider.
A Gates Cambridge scholarship is full funding to receive a Master's or PhD from the University of Cambridge, UK. They're ridiculously competitive, but if you manage to get one, you're essentially guaranteed a PhD from one of the most prestigious universities in the world. From my experience, admission to PhD programs at Cambridge is limited by funding, not space, so for most applicants, they will be admitted to the university but then won't receive funding and will have to withdraw. But anyway, here are some specific tips/thoughts:
1. Competition is ridiculous. I'm not sure how much this reflects the selectivity of Cambridge versus Gates, but I was eliminated pretty much immediately for the Gates. The way it works is that each department in the university ranks all of the applicants and sends information on the 4-6 highest-ranked candidates to the Gates committe. From this, the committee narrows it to around 2-3 for an interview, and then from the interview, it selects half to receive the award. This goes on until about February; after that, Cambridge moves on to the next top third of applicants on the list and considers them for funding. I was apparently ranked in the bottom third and hence asked to withdraw my Cambridge application, haha. Tough crowd! One potential confounding factor is that the advisor I applied to work with accepted a faculty position at another university and would have been unable to advise me anyway... I'd like to think that him leaving influenced the department's decision to rank me so poorly, but I'm not sure.
2. Now that I've established that you have a great shot at the Gates if you do exactly as I did, I'll tell you what I did. I e-mailed a brand-new professor (from now on: Alex) and said I was interested in such and such areas he was researching. Then, he said he had room in the lab and would like to have a research proposal. We e-mailed back and forth and talked on Skype to get a rough 3-year plan. The difficult thing about Cambridge is they want you to have an exact plan on everything you will do during your three years there so that you do, actually, graduate on time. The department I applied to (Experimental Psychology) wanted a polished two-page proposal in the later stages of my application. Surprisingly, Alex helped me go through a few drafts with him before I submitted. The US seems to be a bit unique in that its graduate students arrive, explore their interests, and then work towards a dissertation. From what I've seen, the UK is very much "here's a project I'm working on, it's yours" or "I want to know exactly what your plans are, why you're the most qualified person in the world to do them, and why they're important." So, get a very solid project lined up with a professor who is very interested in you before you apply, or you don't have a good chance at getting funding.
3. If you do find a professor who is interested, and after e-mailing back and forth for a few days/weeks you have a very solid project established (you know exactly the materials you need, the schedule of experiments / fieldwork you'll run, etc.), start working on the essays. Here, much more so than for the Fulbright, show your academic qualifications. The last guy from U of I to get a Gates had eight publications by the time he graduated. Show that you not only have the techniques required but also the mind to carry out the experiment and make everything work. Find very solid reasons for why your project is important. The department needs to rank you high enough for the Gates committee to even read your application, so hit them hard with your research qualifications and rock-solid broader impacts logic, then leave the flowery writing to the Gates essay (not sure actually what consistutes a good Gates essay... the examples on the internet range from talking about the links between the art of painting and molecular biology, Fulbright-style "this is the story of how I came to apply", and hardcore "I have these qualifications. These are what is required to do this project. This is the hole in the world my research is filling.").
4. Once you open a Cambridge application, update it regularly (once every few days), even something meaningless if you have to. If you don't update it for two weeks, the application program sets a timer on your application to terminate it in one week, regardless of how many times you update it after the timer's set. That happened to my friend Beth and really stressed her out, haha.
That's all I can think of right now. I think I didn't get the Gates for a few reasons: my project involved working with birds, and I had no previous experience handling birds. I'd never published before and had worked in an animal behavior lab for only a year by the time I submitted my app. I tried putting a conservation spin on my project (corvid cognition and social learning)... minimal implications for conservation, unfortunately, so I'm pretty sure the department saw right through that (it's worth noting that you can still have very strong 'importance and outreach' implications that don't require conservation). Otherwise, it's probably just a testament to how insane the competition is. I had good grades, extracurriculars, and strong letters of rec but I think research experience is at least 90% of their decision.
Good luck and let me know if you have any questions!
IBH students travel the world, publish research papers, and do all sorts of amazing things