Talk to me about relocating and school applications

Your y10 is the biggest concern here, as they really need to start at the beginning of term and you may have greater trouble with subject options, if some of the options are full. Once you have a property, I would ask what is the earliest you can apply e.g. can you apply with a confirmed flight home and property. You usually have to take any place within 3 weeks, but if you could apply 3 weeks before moving, that would give more time, as once schools shut for summer, not much happens. I would also try to work out which schools have places and include 1 on your list that does. Ideally your DC would start for the very end of yr9 to get into school and choose options. You will probably have to accept whatever school has places for GCSEs and consider moving for sixth form.

With your yr7 you have a bit more time to hold out on waiting lists, especially if you could manage them starting a few weeks into term if needed. There is more likely to be movement in the summer or early Sept when someone doesn't turn up.
Edited to add: As @MarchingFrogs said I would consider doing an application now to some less popular schools to try and 'bank' a yr7 place. Then in June you can change your address and either move up waiting lists or reapply to new schools based on where you end up living.

https://www.gov.uk/guidance/schools-admissions-applications-from-overseas-children

You can apply from abroad- but the address used will be the one you apply from, so the only way you will get an offer straight off is if you have named an undersubscribed school as a preference - without a local address, what will not happen is the LA finding your 'nearest undersubscribed' school to allocate, if none of your preferences can be met.

The in-year admissions is trickier, because you cannot apply now and be offered (if a place is available) and expect the place to be kept open for nearly a year.

Deadline for yr 7 is 31st October

I'm that area @qwert987 and you are likely to definely not get a good choice by then. And I'm afraid that a lot of the schools here start the GCSE curriculum in year 9 so you would need to factor that in to where you apply to. If you want to message me school names I'll see what I know/can find out.

You will get places somewhere, likely to be Quest in Selsdon or Meridien in New Addington. And possibly Harris Purley.

Sadly you're doing this a year too late

I would be handing in my notice and breaking the contract. Unlikely they’ll pursue it!

Do you have jobs lined up here?

Waiting lists are held in order of criteria, not how long you've been on a list, so moving as close as possible to your target school is the best bet.
Theres a good chance the places won't be in the same school.

On the off chance.. are you abroad working for the Government or just private individuals. There is a slightly different process in both cases.

We put my neice on the waitlist and she did get a place for September in a popular city school. However there are 3 schools in my town and there is no chance of getting into one of them on a wait list.
I think it depends on location.

I would move heaven and earth to get a place so you can apply in time

This is going to be stressful OP, as both children will have to go on the waitlist it could very likely get to September and you are given an under subscribed school miles away.