I feel for you. The London school admissions process is a nightmare when you come from abroad - particularly if, like you, you are looking for a place in a year group where all are new starters ie Reception and Y 7. It is easier if you are applying for a place in a year with less movement as more schools have vacancies then as families move away.
You cannot usually apply for a school place unless you have an address in the borough and are living there. And you have in any case missed the closing date for this years allocations so your application will be treated as an ‘in year’ application. This means you will be allocated a place at the nearest school with a place in the relevant year group. Places can be in schools miles away and siblings will not necessarily be allocated a place in the same school which is no fun.
However the good news is that you can put your name down on the waiting list for a place at your preferred school. Your application will then be looked at in accordance with the admissions criteria for that school - not according to the date on which you applied. So, say for example you are allocated a place at a school 5 miles away but there are four closer schools, you need to accept the place at the school offered. But the Council will also accept your waiting applications for the four closer schools. If your house is next door to one of the closer schools and the main admissions criteria is distance from the school, you will be offered the next place that comes up. Your child can then move. This is disruptive but the smaller they are the easier it is. But do bear in mind that there is not much movement in Reception classes.
Some schools have admissions criteria other than distance. Do you belong to a religious denomination? Was your child adopted? Does she have any special educational/ medical needs etc? Because those factors can also be taken into account.
You can also choose to send her to a private nursery for the first term and move her to a state school when you are offered a place you like. Just do not tell them that that is your plan. You will probably have to pay an additional terms fees to the private school if you remove her mid year - so watch the contract. But I suspect that post Covid there will be many empty places in private pre preps next academic year so you may be able to negotiate.
The thing to remember is that it will all work out fine in the end!