I moved to London with 3 primary aged children in the summer. I agree that the first thing you ahve to do is phone the admissions team of the different local authorities. They will tell you which schools currently have places in reception (agree it is unlikely at this stage to be any of the outstanding ones, but who knows, people may move at Christmas!). I would also say you should phone the schools that you like. They have more up to date information than the local authority about the families, and may have heard who is planning to leave at Christmas. they may not have notified the admissions team then.
When you have the information, you should rent a flat as close to the school as you can. You will only get a place at the school (unless it is a church school ) if you are top of the waiting list. For most schools that would mean living closest.
If you are lucky enough to fulfill church school criteria then you may get a place in a church school.
I don't know about the areas you are looking at (although I did phone Wimbledon as part of my initial search), as I moved to N London, but I expect it is quite similar! in N London there aren't places in reception in the good schools, but people do move on alot. I got offered places in a very good school for all 3 of my children (now years 2,4, and 6) which I was amazed at!
As for your final question about going to a private school for a year, I wouldn't do this if funds are tight. I would put your ds into a state school and then move him when places become available in your preferred school. You may have to rent very close to your preferred school to do this (check catchment areas with the Loc Auth), but places should become available in time.
Good luck!
It is very stressful I know.