I've had an old Kindle (4th gen) for some years and also use the Kindle app on my Android phone. On both of those when I purchase a new book or upload a personal document it will trigger an auto-download.
On my new Oasis 2019 I see nothing unless I restart the device. Not even the placeholder in the library, even though I expect it to fully download in the background.
I have set it as my default device, I have reset to factory defaults, disassociated from my account and re-added it. No change.
Network change makes no difference, only a full restart.
OK, I was impatient. It appears that this is a function of the default indexing that goes on in the background - but of course it's not documented anywhere except by users online.
I had sent a large number of documents to the new Kindle and unknown to me until now was the indexing. It seems that downloads are suspended while this is happening but on a restart I guess the download job is kicked off before the indexing starts.
So, in case anyone else ends up reading this, check for the indexing (search for random text from the home page, select search everywhere and then Text In Books if not dimmed and that will show you how many books are still being indexed).
This will almost certainly also explain my battery drain (diff thread).