I would definitely recommend moving out of London, but not too far away.
I absolutely loved living in London and couldn't imagine ever being happy anywhere else. I thrived on its busyness and energy - until I had my first DC and went on maternity leave and then I started to feel differently. By the time I had my second DC (they're just under a year a half apart), London had completely lost its appeal and felt too busy and polluted. I started to feel stressed and anxious all the time, as getting anywhere further than walking distance away was such a hassle with two very small children, even though I was very fortunate to live in a lovely part of London (next to the river in Putney). I always seemed to be either stuck in traffic or trying to get a double buggy on and off the bus. And parking outside my home was always difficult as the residents' permit zone covered about ten streets, so I'd often have to park on the next street and struggle to my front door with shopping bags in one hand and carrying my DD in her car seat in the other, while trying to stop my toddler DS from running into the road.
We moved to a town 20 miles away, that's now on the Elizabeth line. The lifestyle change was unbelievable. We could afford a house with a driveway and a proper garden and not just a maisonette with a tiny dark courtyard. There are so many green spaces and parks everywhere, a choice of lovely primary schools with big playing fields and a proper feeling of living in a community. And in less than half an hour I can be in central London.
I knew no one here when we moved, and it took about six months to feel settled and to stop constantly driving back to Putney to see my old friends. But it's such a friendly place and having young DC made it easy to make friends with local parents ( I volunteered to be on the committee at toddler group, joined the PTA, helped out at Brownies etc.).
Now my DC are late teens and pretty independent, I've rediscovered my love for London and go there often, but it's such an easy train journey I feel like I have the best of both worlds living here and would never want to move back.