Ok I've namechanged because this could be outing, but it's probably worth saying.
As has been touched upon upthread, there is the 1 in 10 gifted child which every class will have one or more of; then there is the 1 in 100 gifted that most schools will have one or more of. The 1 in 1000, a school will rarely have more than one at a time if that. The 1 in 10,000 - will stick out almost anywhere. But if a school is taking a high proportion of 1 in 100 kids, they may be better placed to work out what to do with the 1 in 10,000. A school that is relying on the 1 in 10s just doing fine by themselves, will almost certainly fail that 1 in 10000 child if they come their way.
I was like the OPs son as a child, a 1 in 1000 or 10000 - I don't think anyone at school ever taught me any maths, I could just work ahead through the textbooks. Just after starting school I worked through a series of American primary school maths workbooks I'd seen in a newsagent and asked my mother to get me. Later, I sometimes got enrichment work at school, sometimes I didn't - depending too much on the teacher I had and the degree of personality clash. But I was always expected to do what the class was doing as well and sometimes got very bored with problematic consequences.
Finally at the end of year 10 I was picked up for olympiad training, and less than 18 months later made my national team, and went on to get a bronze medal at the IMO. If the right people had known about me earlier I could certainly have gone further than I did (even though that was literally beyond my wildest dreams). Sometimes I wish I'd moved to the UK for uni - might have got better mentoring possibly - but I was impatient to get on with actually learning something and moving from the southern hemisphere would have enforced a gap year.
I think my own 12 yr old might have the same potential, and the similarities to the OPs son are more marked, he's teaching himself all sorts of things from youtube etc. We are extremely lucky to have him in an excellent small prep that has enriched his education wonderfully; he is about to sit music and academic scholarships for senior schools. The maths ability positively influences a host of other areas. If I could turn the clock back, I would have prepared him much more for pre-tests when he was 10, as he interviewed badly. I'd hoped he would be able to have some control of where he wanted to go, but in retrospect I didn't understand that he had no image of what senior school would be like and until much more recently wasn't ready to think ahead to it. We are now hoping for a boarding place at either a highly academic school or our second choice that we believe will be able to nurture his special abilities.
To the OP - reading this thread I think you have taken on board that you need to be taking charge. Look as widely as you can, ask questions, and don't think any school would be presumptuous to consider. If you can get appropriate maths textbooks that might be helpful for your son to work through things systematically - repetition is key to mastery even for those to whom the concepts are trivially obvious. Depending on where you live, don't rule out boarding at age 13 as a possibility, but now is the time to start looking at such schools as well, confusingly.
When you find the right school environment you may find that your son knows it. Or he may have to grow up a bit more first; trust your own instincts as to what is best for him until he can appreciate it for himself.
Very best wishes and feel free to message me.