You could also send the following letter to the UBA:
In the matter of [Geschäftszeichen], according to my records, no further correspondence from you has been received after [Datum des Schreibens bzgl. der Unvollständigkeit]. If this is different according to your records, please inform me by [angemessene Frist]. Let your lawyer judge whether this is useful to strengthen the presumed approval. Presumably, he will simply advise sending a construction start notification, or better yet, have this done directly by the law firm, then the building authority will know who’s in charge.
We had a similar case where the official called proactively to request an extension of the deadline. Otherwise, he would have had to go to the construction site daily to prevent the construction.
That authorities ask applicants (by phone!) for deadline extensions, I have never heard of. Why should the official want to go to the construction site? – the approval by operation of law takes effect automatically after the deadline expires, done. As far as I know, the authority has no legal remedy against this. If the authority imposed a construction stop solely on the grounds that it would have liked more time, it should be made to pay for its lesson. Anyone with half a brain could handle that more skillfully.