Have you applied to the LA for a place in “any school in the borough except the one your children are currently in,” or are you applying for a specific school!
I assume it is a specific school, so how long it will be will depend on if there are currently places available in the respective year groups. If there are it should happen very quickly, and the school will not want to hold places open so be prepared to move fast. I have known a phone call on Friday to start Monday, which I think is too short personally, means both schools are rushed, especially the receiving school with no time to prepare things to welcome a new child.
If there are no places you will either have to accept another school, go on the waiting list, appeal if it is a Y3 or above, which of course will take longer.
But the fact is some LA are a lot more efficient than others and will turn applications round in a matter of days, while others are a lot less efficient.
( School census day was on the 3rd and returns have to be in by the 30th so you have probably missed those deadlines, which won’t put a smile on the new schools face.)