I am not applying for maths myself, but I can offer some general advice from what I've learnt.
I was emailing a Cambridge lecturer and he stated: "don't worry too much about needing to mention the 'right' things". It doesn't matter if the opening is cliche or original. They're looking for your talent in maths, not in opening paragraphs - use a tinned opener, or just get right into the meat, it won't matter.
There's nothing to be ashamed of in developing your interest late, and there's no need to mention your a 'late' bloomer (and I'm willing to bet the majority of candidates are just the same) in your statement or even make explicit reference to your interest - that you're interested in maths should be patent from what you include. Even then, reading maths books at all - no matter how superficial - marks you as a cut above the rest.
If I may, this is my suggestion for a possible plan:
- What attracts you to maths (if you don't know, it's fine to ignore)
- Talk about reading maths books as a lad
- Make massive mention of your solving STEP problems and olympiad questions in your own time, perhaps referencing particularly complicated ones you solved, and state that what you find best about these is the needed creativity of the solutions - Cambridge is mono-maniac about creative and original minds.
- Mention that you scored silver in UKMT - still very impressive.
- Finish off by discussing how your interest has impelled you to tackle some proper books
- Include a short paragraph of your generally applicable achievements, like sports, if you have any room.