I think it completely comes down to how many kids you want to have- if any. With none, or maybe just the one, Battersea would be better - closer to central london, nice modern apartment with concierge, lots of nice restaurants right in your doorstep- there are a few schools not to far away (newton prep for example). When we just had the one we lived on the southbank and loved it.
However, a bigger family would probably need bigger space and more access to green space/extracurricular classes, etc. if that is your plan then I would move a bit further out….
Hope that helps!