If something existing becomes something new, then the question arises at what point exactly (in a period of time / or space-time) such a transition - from the existing old into a new - happens, doesn't it?

That transition occurs at the exact moment the change occurs. What changes becomes different, and if no change is verified, then it would remain the same, old as before.

Actually the term "novelty" is misleading and one would have to say "otherness", which in turn is linguistically connoted quite differently.
For example, if the virologist says "I have identified a different virus", it would sound less dramatic (or scary) than if he says "I have discovered a completely new virus!

Perhaps the virologist should not say that he found a completely new virus, but he did find a partially new virus, because it is partly different (or new) and partly the same as before (or old); partly it is new because it is not exactly the same as its predecessor; but in part it is the same because it is not completely new but there are parts that remained unchanged that make them like SARS, otherwise, we would not relate them but we would see them as two different viruses.

Therefore, despite being a whole, apparently there are parts that differ in their qualities.

"New" suggests something unknown. Something that we don't know suggests that we don't know how to deal with it. It suggests that this newness does not allow old familiar considerations, is not accessible to common treatments and methods.

Yes. But the same applies to the word "different" or "other", how can you know something that is different from what you know? How can you use the same treatments and methods for something that is completely different?

A counterpart for your thesis would be to say that, although it is true that we cannot know exactly when said transition occurs (from the same to the new), that does not mean that said transition does not occur, we do know that such transition occurs, and it is best to assume that as things are constantly changing, things are constantly being renewed and are permanently new. In this case, the novelty would be the most common thing in the world.