Hello. I so feel your pain. Central London schools are not the easiest to navigate.
First things first: call nursery schools you're interested in asap.
There is sometimes a January intake and getting your eldest in would not only resolve that issue in and of itself, but, because most have sibling policies, square things away for your younger child as well.
Many will be full, most will say they're full, but some will have last minute spaces for children starting now (your 3 y.o.)--places they'll release to you if they sense that you're a good fit. That's true from The Acorn to Strawberry Fields to The Minors and on down the line.
Be polite but persistent. And if you know families at a nursery you really love but can't seem to get into, get them to write on your behalf.
Second: get your children on the wait list for primary schools. Not all schools will have wait lists, but many that you may want will. And there are cutoff dates, so move quickly.
Just go to the school website, print out the forms, then go in person to drop them with the registrar.
Some people will say you're screwed because your children haven't been down from birth. And let's be honest: for a good number of the hyper-selective central London schools people really do register their children as soon as they're born, so coming in this late to the game isn't ideal.
However: the fact is that the vast majority of children who are down from birth still don't get in--whilst others that are put down later do. So let's be clear: none of this is as cut and dried as it seems, and there is nothing one can do (short of having a child already enrolled in a school with a sibling policy) to guarantee a place.
That's good news for you. If you can get your children on the wait lists, you can work to get them in later.
Third: once you've gotten the above out of the way, start visiting the schools that will allow it (some of the most selective won't allow visits until after your child is in/after you're high enough on the wait list that you have a real shot)--both those you've put your children on the list for, and others that may not require anything so extreme.
That's how you'll get a feel for which schools you like, and in what order. You don't need to decide now. You do need to take the steps that will ensure you have an option to pursue your preference once you have one.
Make any sense?
Long post, and maybe too detailed.
I just find it so boring when people tell newcomers horror stories but aren't proactive about helping them get a foot in the door. None of this is rocket science. And though it is stressful, it can be managed if you can keep your cool and stay organised.