As I said in my previous post, once you move and have a UK address, your local authority HAS to find your DC school places. The schools will be where ever has places. These may not be the 'best' schools in the area but you may get lucky.
The school may be some distance away from where you're living (if it's very far, your DC might be entitled to transport but the Yr10 will certainly be expected to travel independently and London transport is pretty good/reliable).
Once you're here, you can also go on the wait list for whichever schools you like the look of. The council (or the school) will then tell you what number you are on the waitlist.
Your choice will be to: send them to whatever school the council has given you and then move them if/when they get offered a place at a 'better' school.
Or you could home ed them if you aren't happy with the schools they have been offered until a place comes up on the waitlist. However this is a big risk as some schools may not have much movement and you could be waiting years. Also, moving a Yr10/11 student isn't really recommended unless you really have to due to different exam boards teaching different topics.
Usually, the closer you live to a school, the better the chance you have of getting in (there are some exceptions). So eg if you found an amazing secondary school you will have a better chance of getting in if you rent somewhere nearby. But! Everyone is aware of this so you often find rental properties near very desirable schools are very expensive.
Also, the local authority only has a legal duty to offer your DC a school place. If you turn that down, they have no obligation to offer you another one so they will have discharged their duty and it will be up to you to provide them with an education.
It really isn't ideal for your YR10 DD to miss the first term but, it does happen. People move all the time in London and schools are used to dealing with it. As others have said, it may be that she's quite 'advanced' educationally anyway so for maths and science she may have already covered the curriculum. With things like English Literature, she will have missed the first set text but you could look at getting her tutors to help her with this. The school will work with you too.
You can google 'GCSE maths papers' to get an idea of what the exams are like. However, to make things even more complicated, there are several exam boards here in the UK and different schools use different boards! AQA and Edexcel are two of the most popular so you could look at those. There are also two levels in Maths GCSE: Higher and Foundation. Higher is for the more able students. So if you google 'GCSE maths AQA higher papers' for example you will get an idea.
I know this must seem like a lot to navigate, especially as a single parent. Schools in London are used to children joining at all sorts of times. We had a girl join my son's primary school in Yr6 for the final six weeks before all the kids left for secondary!