Sounds like you were "lucky". I know someone with a DC, now Y2, who lives in E Dulwich. They filled in all six spaces and didn't get any of their SIX choices. They were allocated a school the other side of Camberwell, up near Oval on the Lambeth boundary. They went private - possibly unsurprisingly. Were your first two choices Heber and Goodrich, by any chance?
PPs saying things like nearest school is 45 mins walk, well, yes, but if you choose to live 45 mins from the NEAREST school then you'd expect that, surely
? It's a rather different situation when you're allocated a school you have to drive past NINETEEN schools to get to, or else get a rush hour train into central London, a tube, then another tube back out to school, then a tube back into central London, a tube back to your usual mainline terminus and a tube onwards to work, like the people we know... In SE London, walking is the fastest route, not the slowest option. I regularly walk a couple of miles with my 4yo and we overtake buses.
On the plus side, OP, there is a lot of movement as a PP has said. Even if there's no place by the start of term, there's a very strong chance something will come up within the first term or two. Depending on the age of your child (they don't have to be in school till the term after their 5th birthday) it may be worth accepting the place, going on the waiting list for all your local schools and simply deferring till the necessary term, in the hope a more convenient place will come up. You'd have to either have a SAHP or a willing nursery, but it's an option. That was our plan B for DC1, knowing that as a summerborn, there was no need to be in school till the following school year, which would have given us a year for a vacancy to come up at just one of our preferred schools...
Good luck, it's a horrible situation to be in.