I live in this area, but have boys rather than girls, so as well as Highgate its schools like The Hall, UCS, Arnold House, etc, that the boys aim for. Same thing as the girls schools you name though.
I agree with PP that people don’t tutor for 4+, BUT they do send their DC to nurseries that really focussing on prepping for 4+, eg Hampstead Hill, Broadhurst, Mulberry House.
My two DS got into “top” boys prep schools at 4+ and as far as I can tell it was because they went in happily when dropped off for the assessments, interacted well with the other kids and played collaboratively eg building a big Brio track, and would listen properly when read a story and then talk about it.
Things to work on at home include reading and writing their own name; fine motor skill things like threading beads and doing up and undoing buttons and zippers; being able to draw a picture of their family (stick figures are fine but teach them to include fingers, toes, noses, ears and eyebrows in the drawings); doing simple jigsaws; talking about the stories you read then (eg which character do you like best and why?, what do you think will happen next?); using scissors; saying please and thank you and a polite hello and goodbye while looking the adult in the eye.
Good luck!