Caveat - we are NOT London. It sounds as if you have a bright, happy child doing plenty of extra curricular stuff. Presumably violin & piano are her instruments (just wondering what else a 4 year old could play)?
DH and I were encouraging our DD(10) at that age. She was a lovely child but didn't seem overly bright (was a summer baby so not as 'clever' as many of her classmates) and the nearby school was great. Something seemed to "click" at the end of year 1/first term year 2. Head and staff were encouraging when we wondered aloud about 7+ and said they thought she would thrive.
We entered her for her local selective 7+ and did a tiny bit of prep (or rather we read the syllabus guidelines and filled in the gaps). We took the view that we didn't want to push her if she would struggle ever after.
We took a similar view this year for seniors.
Music and sport are a core part of the school curriculum. Working hard and 'trying hard' is positively encouraged. Some of the music groups/sport are very selective, others aren't. I suspect this varies significantly from school to school/enthusiasm of staff and facilities and the general ethos of the school (winning at all costs vs having fun).
What made our child a good fit (looking back to age 4-6)?
- a desire to learn (asking questions often based on things that happened a lot earlier). Good vocabulary (use and understanding)
- happy to give things a go (sport/art/music - actually music took a while)
- taking instructions and (generally) following them
- not being too upset if things didn't go as planned
She genuinely loved going to school (up with the larks and determined to be there as early as possible).
When we were looking at schools we were conscious we were uprooting her. Try to find some things about the school you know she will love (even if it is a tennis court, playground layout or a set of books in the library). Take DC to open days (I've probably not seen 4 year olds, but certainly 5 and 6).
If a child likes a school and thinks the teachers are wonderful they are going to be motivated.
A sizeable minority of children seem to swap in and out of the various schools here. Most move to a school that suits them and are a lot happier. I feel far sorrier for the children who don't bounce in and out of school and really don't seem to want to take advantage of the opportunities on offer.
Good luck OP. I'm not sure there is just one "right" answer.