I think that you need to decide how you want to live with v young children in central London....how active, physical or outdoorsie would you like to be with your children?
Do you want to be able to throw open the back door so that they can play for hours safely in their own back garden, climbing trees, getting dirty making mud pies, digging for worms and building dens etc? Do you want them to have a small veg patch or green house to grow veggies? Do you want them to keep pets (rabbits, guinea pigs etc)?
Do you to want ensure enough outside patio space for them to be able to skid around on a tricycle or scooter?
Do you want your girls to have space for a wendy house, a climbing frame, swings, slide, monkey bars, paddling pool etc? Best thing my kids have had is a trampoline - they are on it all year round - takes up a fair bit of space tho.
Do you want to be within a quick (5mins max) safe, non polluted, scooting/ cycling/walking distance to a big park so that you can easily go everyday?
Or do you want it to be all about the house - so directly on to a v busy, noisy, polluted rd, with a long, stressful, noisy, polluted, hazardous walk or bus ride to get to a park for the girls to play?
It really isnt about prioritising chi chi coffee shops and cake - frustrated toddlers strapped in buggies or bored little girls drearily colouring in whilst yummy mummys gossip - childhood should be more than that? Not child centric IMHO.