Maths Olympiad participants, how did you start getting good? Watch
Recommend books, given I am GCSE student. I think you need to know stuff like Chinese Remainder, Pell's equation and Quadratic Reciprocity, Euclidean Geometry, Ceva's theorem, excircles, Menelaus's theorem, power of a point and the radical axis theorem. Give me tips on how to start.
UKMT do books about individual topics (number theory, geometry, and many others), which are good. Just do past questions. I honestly don't know any of the things you listed above to any detail (apart from Euclidean geometry because that's kind of important), and I got a merit in BMO1, so the key is really practice questions, more than learning theorems and equations. Heron's formula and the two trig triangles (30-60-90, and right isosceles) are the only two things beyond basic Year 9 maths that I'd say you'd even need to know, and you can practically get away without know them either.