Mathematics at high school and that at university could strike as two different poles altogether. High school mathematics has a more methodological approach. It teaches you how to apply concepts wherever needed that is sufficient for students who choose a degree that requires the application of mathematics. But for the minority who choose to study mathematics itself, the high school curriculum is not tailored to their needs.
And bridging this gap is quite a challenge for most students choosing the degree. You are no longer using calculus to solve a problem. You are making mathematical justifications in almost a new language, discovering the very principles of each aspect.
The faculty at UCL, however, is supportive and considers these difficulties. They have tutorials in the first year, which comprises of small groups. It gives you more space to articulate your thoughts and takes away the pressure of asking doubts amidst hundreds of students. It adds more clarity to the process, which is especially helpful when you are trying to grasp the very fundamentals of the subject.
Many people, sometimes including maths teachers themselves, discourage the idea of studying the subject at university, as they don't believe there is something to get out of that degree. However, mathematics does not generally close any doors for you in terms of careers. Instead, it gives you the ability to think critically and push yourself beyond your comfort zone. And to expand your options, UCL does offer courses that integrate maths with other subjects, such as management, economics, statistical science, and even languages! There is also an opportunity to take a year off to study abroad.
Just like all universities in the UK, you can apply to UCL through UCAS. While the whole application is direct information you need to fill in, the personal statement is the part you need to plan through. University can get hectic at times, and colleges want to know if you can take it. Through your statement, you are giving them a first-hand insight into your overall personality. They want to get to know you, something apart from your inclination towards mathematics.
University, of course, is an unfamiliar place. Settling in and social interaction can seem difficult for the first few weeks. Things can be more complicated if you happen to be an international student, immersing in a completely new land. The key here is to give yourself the time you need. Do your best but don't beat yourself up if your performances are different from what you expect from yourself. Try to be empathetic towards your circumstances, and things will start falling into place.
You can find more information about the UCL mathematics department and ways to apply here.