Welcome to my insiders guide to getting into UCL (or getting your child into UCL).
Perhaps youre only considering applying for UCL. I can confirm: you may as well firm your choice now. Its consistently in the top 10 universities in the world. Its second in the UK for research (these world class researchers will be your teachers). And its in the best city in the world. No university can quite compete with our notable graduates:
Mahatma Ghandi
Coldplay
Christopher Nolan (produced the Batman trilogy, the good one, not the upcoming Affleck ones)
To name a few!
There are a million and one other reasons to pick UCL but Ill cease the endless plugging for now.
This article should outline the basic procedures for admission to most courses at UCL, as well as deliver helpful tips for excelling at every stage of the application.

Picking a course:

Above anything else I may tell you, this decision has to be your own and it has to be based on what you enjoy and founded on solid research. You have to study it for 3, 4, 5, even 6 years, youd better make sure its something in which youre fairly interested.

A little story:

I walked into my first year studying Chemical Engineering, I made some friends, I had some fun, I enjoyed the course, and around March people started to disappear. It was eerie, like a horror film, like a demon was picking people off and hoping nobody would notice. This demon is called lack of research! Coming into my second year I think we lost maybe 30 students, they either switched courses within UCL or transferred to different universities. Now I couldnt understand this, I was having a great time, the course was very interesting to me, but then, it was exactly what I expected (and better), after talking to several of my friends that left, I discovered ChemEng wasnt what they were expecting.

Theres nothing wrong with changing your mind, but youll find its easier and more practical to do it before you have to start. So research, research, research!
There are 11 faculties at UCL, each with a range of degree programmes associated with them, it adds up to some 200+ degree courses, and there are new ones being developed all the time! Select one for you and youll enjoy it, and if you enjoy it, youll succeed.

Which A-Levels should I choose?

Which A-Levels should I choose? Always a query on those fantastic message boards (more on these soon). Here I could say pick what you enjoy, but the ugly truth is that universities, including UCL have an approved list of A-Level subjects that they consider the right level of difficulty or relevance for any prospective student (Ill add a link to this list at the bottom). Dont panic, UCL allow you to have 1 of your at least 3 full A-Levels to be non-approved, and your choices are probably all on the list anyway.
That being said, as well as the official approved list, there exists a fabled unofficial hard list among the elite universities. A-Levels they consider more highly than others due to their difficulty, its rather snobby since theyre all traditional subjects but thats the way it is:

Dont be too thrown off by the Hard List, youll likely be wanting to choose one of them anyway. Try to have at least one of them
Apart from being on the approved list, each department requests their compulsory subjects, for instance, Maths A-Level for all the Engineering disciplines (and of course Mathematics), Chemistry and Biology for Medicine, and so on. They may also specify in which subject the department demands an A*.
What about General Studies/Critical Thinking?

Bad news. UCL dont even recognise them as A-Levels. All that means is that you cant brandish your A* in the face of an admissions tutor though. If you have the time and inclination for any of them, they are worth doing, they will hone your research skills and, well, your critical thinking skills, and theyre not too bad to talk about on your Personal Statement.

Insider Tips:

From what I understand, while theres no disadvantage in taking Law for an A-Level, UCL, and most of the top universities dont require you to take it as the course is essentially re-taught in lectures, and they certainly wouldnt like it as a substitute for English or History for instance.

Dont take more than you can manage, better to achieve high grades in 3 than mediocre grades in 4 or 5
Check your chosen departments website to view their Year Abroad opportunities, if youre really interested in studying in a certain country, UCL insist you can demonstrate relative proficiency in the language, and an A-Level in it is usually enough.

Your collection of subjects should be focused but, also wide ranging, pick three very relevant to your chosen degree path, and one not so relevant, or two and one, for instance, I studied Maths, Chemistry, Physics and Geography, for Chemical Engineering. This is also for your own sanity, subjects too similar can get repetitive and boring.

Seek feedback from recent graduates:

UniStats is a site where students of a university assess their experience, it can help you decide if a degree course is right for you, straight from the people whove lived it. However, courses can change, UCL in particular listen to feedback exceptionally well and improve where they can, for instance, the entire Engineering Faculty at UCL have added a component called the Integrated Engineering Programme (IEP) where students from different disciplines work closely together on some projects throughout the year, this debuted for 2014 starters so anyone graduating from a 4 year in 2017 or earlier hasnt experienced this revamp.
Find out if your course has changed recently.
This also goes for new facilities being installed. Currently, development is being conducted on a new UCL campus in the Queen Elizabeth Olympic Park called UCL East. It promises to bring state of the art learning facilities and a new vision for university study. Its all very vague at the moment, but one of our projects last year was to design a pilot plant for the campus, so no guarantees but Chemical Engineering prospective students for 2019 entry may get to play with some new toys!
Essentially, the university is constantly improving, so when conducting your research, take outdated information with a pinch of salt.

Application process:

At last, youve chosen your subjects for A-Level and have considered your choice of course, what now?
Write your Personal Statement
Write your tutors statement (hold on this sounds like fraud all will be revealed later)
Sit any external examinations you must (Medicine and Law students only)
Attend a departmental open day/interview (not always necessary)

Personal Statement:

UCL take a holistic approach to admitting students. This means they, more than any other university, consider your whole picture rather than just one aspect such as your grades. Their first view of your whole picture is from your PS.

Structure of Personal Statement:

Essentially, the key ideas you must include are:
Your A-Level subjects, why you selected them, what you enjoy about them, how they are relevant to the degree you hope to pursue
Your ambition and reason to study the course you chose to apply for, what do you hope to accomplish. Here you should project your passion off the page, show that youve made an informed decision, discuss challenges facing this field in the future that you would like to be a part of
Extra-curricular activities, the relevant ones to your course, have you any achievements or awards outside of school? Have you occupied any leadership roles at school e.g. Head Boy/Girl? Have you attended any conferences or competitions related to your course? Some examples of these could be:
Maths Challenge
Art displayed in a gallery
Model UN conferences
Hobbies and interests; they show youre a human, any sports you play, musical instruments, book genre you like to read, journals you like to read. Competitions, or other accreditation here goes a long way also
These are the 4 musts in the statement, I would suggest structuring it into 4 paragraphs following those themes, but sometimes your skills dont strictly fit this so feel free to play with it, but make sure you have mentioned these essential ideas.

Insider Tips:

Plant land mines: land mines are topics you want to be asked at an interview, so you mention you particularly enjoy reading about developments in cold fusion for instance, study it relatively well. When the interview comes, they expect people have lied on their PS about the things theyre interested in and will try to grill you, when you repel their best efforts with grade A* knowledge you appear calm under pressure, well read, and someone with more integrity than lying on a PS

Recognise the angels from the devils online: The Student Room can be a valuable resource for specific information for your course, but its filled with as many trolls as it is truly helpful people. Take everything from it with a pinch of salt, and dont despair because Malcolm0102 has 16 A-Levels, climbed Everest and rescued the Pope from a burning building but doesnt think he has enough for Cambridge.

Dont overdo it: nothing looks more ridiculous and unprofessional than someone whos made their whole PS rhyme!

Writing your tutor/teacher statement + predicted grades:
If you dont know what this is yet, its ultimately your reference from past employer to your new employer, in this case your old tutor to your soon to be new tutor. It contains your predicted grades, and the tutors impression of you coming from each of your teachers.
You dont really write this, but you influence it, in the end its all about you, and if you can present yourself as an engaging, intelligent and inquisitive individual to your teachers, they will convey that to UCL (and your other choices of institution). You will need to keep this up all through Year 12 and half of Year 13 to fix that impression in your teachers minds so you can get a very favourable recommendation.
In addition, take every test seriously, whether small or large, your achievement in GCSE level doesnt matter too much to UCL nor does it to your A-Level teachers, particularly if youve been performing well on all the tests theyve throw at you. The teachers dont guess your predicted grade, they look at the evidence, make sure that evidence shows excellence.

Insider Tips:

Dont pester: youll usually be told your predicted grades, less likely to be shown the teacher statement, but you endear yourself to anyone who matters by annoying them, youll get a fair assessment, and usually quite favourable, they never want to portray you badly so dont fret about it
This is one of the most important pieces of evidence UCL use to select new students: your predicted grades obviously are the most important here, if they dont stack up, youll be swept away in the first round of screening, if theyre exceptional, you may get a very quick offer. The initial screening takes place early, eliminating any they believe to be not academically capable, make sure youre on the correct side of the broom. I know for a fact that for some courses they look only at your predicted grades and your statement to determine English skills and temperament, then simply give the offer or not, make it count.

External examinations (relevant only to Law and Medicine prospective students):
The only course at UCL that require additional tests are Medicine and Law, requiring you to take (and pass) the BMAT and LNAT respectively. They are designed to test your suitability for each of the courses. There is a cut off for each and each university has its own; they need not disclose it, consider the following graph as a guideline, it is interesting in 2 ways: first, it shows a clear cut off point, anyone scoring below 50% appears to be eliminated (dont panic if you did, this is a guideline only); and that even those who were between 50-55%, about a third got an interview, this is down to their other merits. UCL and other institutions arent barbaric, they know some people have off days and your other academic record should vouch for you if you do.

The LNAT has a multiple choice stage and an essay stage. The multiple choice stage is about reading a passage of content and answering a question about it. From what I have heard from friends that do Law, the LNAT can sometimes seem to have trick questions, it doesnt, only difficult questions, the answer is always there in the text, they are testing your ability to distinguish between two answers that may seem equally right but arent. The essay stage is marked mostly for your English skills and ability to argue a position with conviction.

The BMAT is similar, with 2 multiple choice sections, one on scientific knowledge up to GCSE level, and the other on reasoning, both can be mastered somewhat with practice, but having seen questions for myself, they can be very difficult, dont assume you can get them all right, just answer as many as you can in the time allotted and stay calm. The final stage is an essay question that tests much the same way the LNAT does but is weighted much less with regards to your score than the LNAT.

Insider Tips:

Practice: that is all.

Departmental Open Day/Interview:
Each department does these differently. Usually, its just a tour of the facilities and you can bring parents, sometimes it includes a group activity, sometimes its the tour and a solo interview, sometimes its all three.

Sometimes your offer is conditional only upon your attending so they can meet you. Sometimes its about your performance. They will let you know which is which. For Chemical Engineering, everyone I have met was invited to an Open Day but it wasnt compulsory and we all got the offer without having to interview. For Art students a portfolio must be delivered, and then an interview must be sat. For Law an interview is usually always sat, the same for Medicine. Therefore, my interview advice will be relatively general.

Organise your thoughts so your points come out coherently.
When solving a problem, think out loud for the interviewer, they want to assess your thought process rather than your arriving at the final answer.
Dont try to squirm your way through a hole you wont fit through; dont try to lie about something you dont know about or havent read, apologise and say you havent studied this yet, and then perhaps ask a question about the topic to convey interest, or if its a problem, give any thoughts you might have about how you might solve it.

Try to keep your smile up, responding to high pressure situations is also being assessed.

Insider Tips:

Take a role: in a group activity, you are being assessed, work as a team and let everyone say their piece, this can be dangerous water, the easiest way to come off well in these is to volunteer to keep time (theres always a time limit), this shows you are conscious of the limit, can think ahead and arent shy to take on a role. An observer will give you points, while your teammates bicker about whose idea is best and who can speak the loudest (I am of course joking, make sure your team performs well!)

Be friendly: if you can find an appropriate way to break some ice between you and the interviewer, perhaps you both play a sport, it is worth making friends with them, even a joke, but dont force this, every interviewer is different

Final Thoughts:
I am ecstatic that you have read this far, I hope you have found this guide useful, and best of luck in your applications!

Additional resources:

How to Get Into UCL
Is a London University Right for Me?
Top 10 Interview Tips