I would strongly suggest you get professional advice GABBITAS or the Good Schools Guide consultancy are good people to talk to.
The English education system works differently from that in Australia. Like most (if not all) school systems in the Northern Hemisphere, the school year goes from September to July. In England the cut off point for a new cohort is the 1st of September.
As children start in reception in the year following their 4th birthday, and from Reception then go into year 1, after their 5th birthday; your son would be in Year 9 now or when you arrive in January/February.
For private schools the big intakes are for years 7 and 9. They may allow some flexibility of which year group to place your son, but do not rely on this.
Next London is a very busy place, and whilst schools may try to accommodate you, there is no guarantee that they will have room (even in the private system). There will also be entry tests, which may be for selection as much as placement.
The main GCSE years in England are years 10&11. Then there are A'levels in years 12&13. If you are moving in January you will need to stay for at least 4 1/2 years for him to complete his schooling, or get him into a school which has boarding places, for him to complete his A'levels.
You are also leaving things quite late, it may not be you choosing a school as much as you finding a suitable place. This is why a specialist consultant could help you immensely, especially as you are on such a different time zone.