This all sounds so familiar! I too am a solicitor working in London. Earlier today I had a conversation with two friends, also solicitors in London, about how incompatible law (particularly city law firms) is with life, particularly for women.
First, hang in there! It will get easier. My children are older than yours, 11, 10 and 6, like you we have school fees x3 and a large mortgage but we no longer have a nanny or any childcare to pay for which is a huge burden lifted.
I did take a step back, I am in industry now working as GC. I don’t earn as much as I could but the work is generally interesting and the people lovely. I won’t lie, there are times when I am really busy however I have total flexibility, can generally do school drop offs and pick ups and attend school events.
Transactional work is simply not the place to be with a young family - unless you are willing to accept being reliant on a nanny and disrupted home life / holidays.
I really recommend considering in house, my experience has been that there is a good quality of work and far greater autonomy. I suspect it may well give you the balance you clearly need. Good luck x