TfL journey planner is your friend. Depending on childcare or arrangements you would have for getting a child to school, its not about distance but transport links.
And also about the child themselves. So if your child is confident and energetic, and you live near the 344 bus route, Newton Prep, with lots of outside space, might be an option. It always had a handful of SE1 kids. Or you could consider some of the Dulwich Preps (JAPs, Dulwich Prep London, Alleyns) if you can access them by train.
Are you eligible for church schools? Plenty from south of the river go to St Peter's in Eaton Square and there was another in Covent Garden that used to be popular. That said several from my way also used to commute into Southwark for schools like Cathedral.
In terms of the smaller central private schools, St Pauls Cathedral School, Charterhouse Sq and Dallingtons are well established.
However it only really matters if you plan to stay. If you don't you might be better off choosing your nicest and most convenient school and having local playdates. And if you change your mind it does not matter. There is a steady movement out of Central London and a gradual switch from state to private, so at around 7/8 it is relatively easy to get a place at some of the most sought after state schools (even church ones), places open up at 13+ schools like Newton, and other schools start (Dulwich College, Westminster Under) or have entry points.
Stay! My kids have just finished school but enjoyed their London childhoods. The Primary years were great, with visits to museums, parks, Unicorn Theatre, Borough Market, and lots going on on the South Bank. We had a bit of commuting out for sports clubs, though the same would not have ben true for music or drama, but as older teens they really enjoyed the freedom good public transport brings. Inter alia look out for classes at the National Centre for Circus Arts in Hoxton, which combine hand-eye, balance and other skills and which are great fun.