For any mathematics problem, we can never know whether studying it is wasted time.

For example, look at number theory. For centuries, it was without utility. Then, suddenly, it got practical applications, making all the time spent on it wasted time :-)

Similarly, the Collatz problem may seem useless enough, but what if, in a few millenia, someone applies it to physics or to sociology?

Your analogy is off. With number theory, even if no one had an external use for it, progress was (is) made, new theorems were proved, so time was not wasted in that regard. The problem with the Collatz conjecture is whether the effort spent on it is actually generating any insights into the problem.

But there is 'progress'. Looking at the Wikipedia page:

"The proof of the conjecture can indirectly be done by proving the following:

- no infinite divergent trajectory occurs

- no cycle occurs

thus all numbers have a trajectory down to 1.

In 1977, R. Steiner, and in 2000 and 2002, J. Simons and B. de Weger (based on Steiner's work), proved the nonexistence of certain types of cycles."

I am placing 'progress' in quotes because one cannot measure progress in maths. Before one has a proof, we cannot know whether existing approaches are true dead ends or whether they just need that one extra insight.

That's a valid point.

However, everyone whose work does not directly or indirectly lead to an applicable (or otherwise useful) result has been wasting their time.

Although I guess that's true for anything.

Destinations are not always more important than the journey that leads to them.