From when is it possible to register a change of address?

Please google the words Elternwille, Schulgesetz, and your federal state. If you find anything, then write to the school authority with a copy to the formally responsible school that it is your declared parental will that the child be enrolled in the school at the place of construction. Further justify that you are already working there and that the child has attended kindergarten there. You also commit to ensuring that there will be no non-illness-related absences.
The receiving school has declared that there is space available for the child.
If this is not possible, your child will stay at home until the move. But you actually don't want that at all.

Important: Write the letter very friendly but clearly formulated and send it by registered mail with return receipt.

You will be invited to a meeting at the school authority very quickly, where you explain this again in a friendly but firm manner and do not let yourself be dissuaded.

The arguments must be repeated like a mantra.
And the phrase "our declared parental will" in every second sentence.

So far, this has always worked.

Important: very friendly but tough on the matter.

What does your school law (a matter for the federal states) say about parental will?

we are currently planning our house construction (start of construction hopefully July, August) and have a small problem. Our child will start school this September - but unfortunately not at our future place of residence but 30 km away (current residence)