Well, I figured it out myself.
This seems to be a behavior that is expected. Like it is stated from ONLYOFFICE here:
Due to the operational characteristic, Document Server saves a document only after the document has been closed by all the users who edited it.
This behavior is especially tricky when you use the external storage feature. In my situation, I mounted some Windows network shares to the users.
This means, that the users should always close their document-tabs in order to save the new contents. And to let other users (coming from the filesystem/network-share) see the new contents.
Hope this helps someone. At least I was not aware of this behavior of Onlyoffice.
Or do you know a workaround, to let Onlyoffice save the documents automatically every 5 minutes?