Since after Hilbert it isn't possible anymore to have knowledge in everything going on in mathematics. For most if not all mathematicians today there are many topics in mathematics completely unknown to them. Even if one only focuses on one specific topic, one needs to do many years of study to start working on own contributions.
What can be done against this? Since mathematics is further growing, will it take more and more time for students to study before doing own research?
In some ways that is true. But in a longer perspective remarkable things happen. The Greeks and Romans had number systems in which modern arithmetic taught to children was next to impossible - the fact that the Greeks were able to do so much mathematics without is really astonishing. I am not sure I understand how the Romans did arithmetic at all.
Slightly older students are taught about cartesian co-ordinates in the plane which are so powerful that Euclidean Plane Geometry is pretty much a mathematical speciality rather than a practical tool.
Vectors and vector analysis transform things like Maxwell's equations into an essential and more accessible core.
I think the greater challenge is the level of abstraction now involved, which requires the formation of mathematical imagination [though this is hugely evident in Euclid, for example]. The abstractions are different in different disciplines. Each of the examples I have given involves the creation of a new and unifying language in which concepts can be expressed and understood.
And of course there is also the question of the number of mathematicians, and the effect of the need for each to make a mark - there is lots of mathematics being done, and that affects how much any individual can know.