@Teapotkitten Just to add to the helpful posts you've had today i thought it might be useful to set out the different options there might be. This will by no means be exhaustive but hopefully a bit of a starting point re Sat options:
Firstly as noted there are the conservatoires. These are university level specialist music schools. For example Royal Academy, Guildhall, Birmingham Music Conservatoire etc. They generally have a junior department (or 'JD' where they take school age kids on a Sat and do different music activities depending on the childs specialism. As noted, these places are generally highly competitive. They publish a minimum standard they'd expect but for popular instruments like piano the actual standard they require is often much higher than the quoted minimum.
Secondly there are advanced training centres (I think this is right) - so CYM in London would fall into this category and the other Guildhall Young Artists centres (different from Guildhall in London) such as Norfolk etc . Again there are usually auditions (unless your child starts on one of the primary programmes) and they generally have a minimum standard, but look for potential too.
Lots of local councils have some kind of music service. Ours isn't great, but our neighbouring boroughs ones are really good and often you don't have to live in the council area, though priority could be given to those who do. Locally ours run lessons, ensembles, choirs etc.
Finally some independent schools run their own Saturday schools - one locally to us does and its where my DD does her piano lessons, but also a strings ensemble, choir and piano trio. It all depends on whats feasible locally and we're in London so there are lots of options within a closer radius than in the rest of the country (though the competition for places is insane!). There are also some music schools who confusingly call themselves conservatoires, but aren't and just offer lessons and ensembles - like Blackheath but some combination of these terms and your local area in google might yield some good results.
If composition is of particular interest then it could be worth looking into what options there are locally in any music provision for tuition on this. Also worth considering a second orchestral instrument if of interest - piano can get lonely, though we've found duets and trios to be a great addition.