Thank you for your reply.
I followed the steps in the link you suggested. First, under APNs I did the reset to default. Then, I deleted the APN, and manually setup a new APN, following the steps I found for my device on the website of my provider, including shutting down and restarting the phone
This helped partly. Now, I can finally download and upload files in the Chrome browser (probably also in the other browser, but did not check that).
Unfortunately, within apps I still cannot upload/download files. In Google drive, the setting to only transfer files over a WiFi connection is switched off (but it acts as if this setting is switched on), and in Gmail I don't see such a setting. I don't have any other apps to up-/download files with.
These two apps were the main reason why I wanted to solve this problem. But at least now I can use Gmail and Google Drive within the browser to transfer files over mobile network, which is something, at least (albeit a bit more of a hassle).
Also, just now, I cleared the cache of the Download Manager, and disabled it and then enabled it, as per some advice I read elsewhere, but that did not help.
Since the problem has now changed, there might be other possible solutions to be found on the internet, and to try. Perhaps, you have some more suggestions. When later I have time, I will try again to completely solve this problem.
For now, thank you for getting me halfway there!