It has to run out of all the water that was sitting still between the output hole and the input source where it's heated. If your apartment has a central water tank, that has its advantages - e.g. a whole family inside your place could take successive showers without ever running out of hot water - but this is the primary disadvantage to that, the wait to go through that cold water, which gets worse the further/higher up you are away from it.

Maybe try to find a way to not entirely waste it - like put some of that into a water kettle to be boiled for a hot drink? I enjoy such thoughts but do as you please ofc.

I am no plumber, just thought this might be interesting to know:-). I am 10,001% certain that someone will correct me here if I have said anything inaccurate. 😂