The school year runs from September here: so I'm assuming that you're looking for an in-year place in year 9 (girl) and year 7 (boy) (unless one of them has a birthday before 1 Sept, as that would put them a year above).
You cannot apply for a state school place until you are resident in UK. As year 7 is a main admissions point, there may still be some shake down in who turns up where in September, but any spare places in the more popular schools will be snapped up. What is left by January, and in year 9, is likely to be in a school you would not necessarily choose. If you email and find out what, if any, schools in the area are undersubscribed, you can then look at their OFSTED report (should be linked to school website) to see what the school is like. No guarantee it'll still have places when you arrive, though.
In the private sector, you are mildly unlucky, in that year 7 and year 9 are main entry points in the independent sector and the admissions round has finished. But there is always some churn in London, and schools can quite often fit in extra pupils. The entry for UK students is competitive and exam based, but the school has the freedom to use whatever criteria it likes for ad hoc entry. You will need to ask.
Do you want co-ed, or would you consider single sex. Wimbledon High is an all girls GDST school you might like to consider. Also, the Dulwich foundation schools run an extensive Bus network across south London, so it might be possible to try JAGS, Dulwich College and Alleyns. The schools in Croydon, like Whitgift, might also be a possibility. Ores to look at might include Emanuel and the Harrodian.
It really might come down to which school/s have places, rather than a hunt for a perfect match, and which have a more open-minded attitude to a non-standard application.