KDP,
I don't want to be too down but it could be very very difficult for you to get your children into the school of your choice, especially both of them.
I'm not sure of the ages of your girls, but the the infants (equivalent of Pre-k to Grade 1) are quite full in London at the moment so there's not much spare capacity in general.
The problem is that however close you may live to a school (or whatever other criteria you may fill), that will only put you at the top of the waiting list - you may just have to sit tight and accept a school that's either far away or not very good (or not perceived to be good)- or both.
We moved from the US two years ago and that's what happened to us. We did eventually get a place at the school we wanted, but dd1 has moved schools twice since we came here. Not great considering she was dealing with the whole move from the US to the UK as well.
NOW we have a school that we're happy with - but it's really not been easy. The system here is awful for people moving from outside the area and even worse for people moving from abroad.
Another BIG difference between the US and the UK is that faith schools are also state schools. I don't know what religion you are, if at all, but if you want one of the faith schools then possibly get in touch with the school now to see how you can fulfil their religious practice requirements from the US before you move (when it may be too late).
Good luck!!!