When do you plan to move, and is your 13 year old 14 this year or next?
I ask as if she turns 14 before September then she'll be starting year 10 - which is the begining of GCSEs - in September
But if she's still 13 in September then she'll be going into year 9 and so not he starting GCSEs yet.
If I was moving at this age, I'd want to minimise the disruption in GCSE years if possible, so maybe consider renting first so you can get into the school you want rather than waiting to find the perfect house to buy. (Assuming you are thinking of buying?)
You need your home to be your primary residence (i.e. be registered for paying council tax there) to be within catchment for the school.
People in the UK can't just rent a house next to a school they like while they own a house elsewhere (although people do try!)
But as you're moving from abroad you do have the option open to you as it'd be your only UK residence, so you could use this to your advantage by renting a house really near to the school you want.
When it comes to getting into a state school outside of the usual admissions process, (which you will be as kids usually put their name down for the school they want towards the beginning of year 6) then, if the school you want is full, you have to go on a waiting list, which is ordered by priority, not how long you've been on the list. Distance to the school is one of the major factors so it's really important.
It could be a good idea to rent anyway, to get to know the place first before committing to buying. We did this when we moved to a new town and I'm glad we did as we wouldn't have known about the lovely part of town we ended up buying in.
A downside if this plan though is rents are really expensive just now - it's cheaper in terms of monthly payments to pay a mortgage. However if you can afford to do so, it could work well for you.
Regarding your 11 year old, you've just missed the standard admissions process for him, but if he's turning 12 after September, and so about to start year 7 (the first year at secondary) and if you're moving before September then he will be able to apply as a late admission. Basically there's quite a bit of movement in schools at the beginning of secondary and there's a better chance of getting into the school you want then. Once you have one child at a school, for most schools that makes it easier to get the other child in, as for most (not all) state schools having a sibling there puts you to near the top of the waiting list, if there is one.
Good luck for the move OP! There are lots of lovely places to live here, and some stunning countryside, I hope you enjoy your UK adventure.