The problem about one of you coming earlier to apply for schools is that if you apply and are offered a place/s the schools will expect you to take up the places within a couple of weeks, they won’t hang on to them for you. It would be a good idea to have an idea of where you want to live, and for one of you to come and secure a house/flat in that area, before the children come, because you will need an UK address for your school application. Most applications are made through what is called the Local Authority, which is the administrative office for the area you will be living in and which you can google. Some state funded schools, called Academies, administer their own admissions.
So 1) decide where you want to live and research schools in that area. you can ask the school and the local authority which schools have vacancies in the appropriate years. But be aware that this can change very rapidly. Because of the way boundaries are drawn you might find that the schools you prefer are in a neighbouring authority, don’t worry, you can still apply to them.
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rent accomodation in that area, as close to the preferred school/s as you can , not that it will make a difference to your application, but it willmake it easier to get the children to and from school.
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make the application when you know the children are coming over. They will either be offered a place which you will need to take up quickly, or they will be refused.
if refused you have several options.
Either :Go on the waiting list for the school you want but accept a place at another school
Or : Appeal against the decision ( come back to Mumsnet if you need advice on appeals , they are complicated things)
If there are absolutely no places within a reasonable distance of where you will be living you can apply for a place through what is called “Fair Access”, again , this is complicated , so if you need to come back and ask.
Good luck.