I'd contact several nurseries as soon as possible, as you might well end up on a waiting list.
Not many nurseries near Sydenham station, unfortunately. Cuddly Bears is reasonably close, which everyone seems to love; little walk up the hill. There's a place called Cherish on Jew's Walk too, which I visited and didn't like much, but I can't remember why, and of course it was a while ago now – so it's well worth your visiting it. Cornerstone would be the next closest, I think. That's on Trewsbury Road. A friend's son goes there, and they are very happy with it.
Crystal Day Nursery would be just about doable, although it's much nearer Penge East (don't suppose you can commute via Penge East by any chance?)
Lower Sydenham has a few, including Little Elms, which is where my son went. Everywhere else had quite long waiting lists when we moved to the area (he was about 2 and a half at the time). But he actually really enjoyed his time there and I liked all the staff and management. He's still very good friends with several of the kids from there (one or two have gone with him to St Bartholomew's, which is where he's now at school). When my partner did drop-off she then commuted via Lower Sydenham station; wasn't all that convenient but it was doable.