In regard sputum tests this is something we know about a bit. My daughter has cystic fibrosis and quite regularly has to have sputum samples taken. In the past they did nasal swabs when she was small but they really were not that great. And they were looking for bacteria which are much bigger and more robust.
Once she got old enough to do a form a chest clearance called autogenic drainage, which allows you to shift stuff from deep in the lungs, the samples are much better, but they can also miss stuff.
One time they hadnt found anything although she was very ill, and it wasnt until they did a lung flush that they found stuff dislodged that was really deep in her lungs.
However I dont know how this would relate to this virus as although it appears to attach to lung tissue, Im assuming they are looking for it being shed as it actively takes over cells, and being caught up in mucus within the lungs. From the progression of sore throat to upper respiratory pain, Im assuming its progressing downwards as it goes, so depending upon where it has progressed to, may prove easier or more difficult for the 'shedding' to be picked up?
As an aside, Kings in my daughters specialist hospital and I know her unit has been taken over as a specific Corvid ward as has state of the art respiratory equipment