To summarize my case:
On April 12th, the application arrived by mail at the building authority.
Pretty much exactly 12 days later, the acknowledgment of receipt came with information about what was missing. I collected this and resubmitted it by mail a few days later, including the neighbors' signatures.
A few days later, I was already able to see in the municipality's council information system that my project will be on the agenda tomorrow. However, it was already visible as approved in the public system.
So it seems to be just a formality now that the building permit will be granted to me. The water engineering company (drinking water) commissioned by the municipality has already contacted me, and the discharge permit also arrived at the end of last week.
Therefore, I can say that the processing here took about 4-5 weeks.