Hi,
Applications for state school are based on your address. You will need a confirmed address in order to be able to apply - in some places, you will need to have already moved in. When a school place is offered, you are expected to take it up within days, or it may be offered to another child.
It is true that all 'desirable' state schools in London will be full, and you can only hope to get in by waiting list.
Once you have moved/have a confirmed address, you apply for a place, and the LEA offers you one - they are obliged to find you a place. However this place could well be in a random school that is generally seen as undesirable (hence places available) and it could be quite a distance away, requiring your child to travel for e.g. 45 minutes on public buses or such. In which case you'd probably decline the place. That means that the LEA is no longer obliged to find you a place. But at the same time you apply to the schools you wish your child to attend directly, and go onto their waiting lists. Your place on the waiting list is determined by how well you fulfil their admissions criteria. (If you are religious and would consider a faith school, it would be a good idea to find out now what faith criteria some of the faith schools in your chosen area apply. They may need a letter signed by your priest, or weekly signed church attendance slips, so it would be good to get started in collecting any such 'evidence'.)
Meanwhile, you HE your child.
Now in September 2017 (presumably) your eldest would be moving into Y3 which is Junior School (YR-Y2 are called Infant School). Some schools are through primaries, YR-Y6, but other schools are just Infant Schools or just Junior Schools (Y3-Y6). Junior Schools obviously have their main intake at Y3. Applications close on the 15th January 2017 at which point you need to be resident at the address you wish to apply from. So that would be a point to keep in mind! If you manage to have moved by then, your application to Junior School would be on equal footing as everybody else's (except where Junior Schools have named feeder Infant Schools and give first priority to children who have attended those schools).
Also it may be useful to know that in Infant School, YR-Y2, schools CANNOT (except in rare, exceptional circumstances) take more than 30 children per class - so when the school is full, it's full. However from Y3 onwards, they can and do take more, if space allows. So from next September, when your eldest would start Y3, you have much improved chances of getting into a school. The process then is that you apply for a place, are rejected due to being full, then you go to appeal.
One thing to perhaps consider: If you can afford to pay £3000 in rent, you could also consider living somewhere cheaper and paying part of that money for private school fees. There are some American Schools outside of London but within commuting distance. That would be something I'd consider, especially seeing as you are moving back to the US system after a quite short time.
Regarding your younger child, she qualifies for 15 hours free 'childcare' from the term after her 3rd birthday. So if your child was born Jan-Mar, it starts in summer term, if her birthday is April-Aug, it starts in Autumn term, and if it is Sept-Dec, it starts in Winter term. I think it is 30h if both parents work, or will be soon. These 15 hours can be taken up in private day nurseries, private or state nursery schools, or at qualified childminders. You can choose to send your child for fewer hours, or for more and pay for the difference. With childminders and private nurseries/nursery schools, you may be able to send your child sooner, but you will have to pay the fees up until your free funding kicks in.
Applications to these places are usually directly to the nursery/nursery school/childminder. You don't have to have moved already in order to secure a place, but you may have to pay a deposit (e.g. £50). Some will be flexible and take your child for e.g. 2 whole days (9-3) and a morning, others will have set times e.g. 5x 9-12.