I don't know if this would help. I am from China. My dad went through the same thing. The sputum test was done at a designated hospital in November 2019. All information was sent from the panel physician to the hospital before my dad went for the test. He only need to mention his name and show the passport during the test and the result was sent back 8 weeks later to the panel physician directly. My dad actually got a phone call from the panel physician for the result, not the hospital who did the test (which is different from your case).
My dad was supposed to do another X-ray 3 months after the first medical test (in Feb 2020) required by the Panel physician. It got delayed as the panel physician office got closed due to Coronavirus Outbreak, but the result was uploaded within two days of the X-ray though (X-ray March 2, passed March 4). I don't think you need to worry about uploading anything based on my experience.
Still IP, no DM, no PPR since then.
Hope this can help. Good luck and stay safe.
Your detailed explanation is of help to me as my mother faces the same situation. She is in India. She did the sputum test early February and results awaited. With the whole COVID situation, I was wondering what will happen once the results become available in 8 weeks as she will need to do a follow up X-ray. Right now, everything is locked down in India for 21 days. So I am sure the medical process will be delayed which is OK and expected based on the current scenario everywhere.