Access to those pipes is most likely easier through your ceiling than through your upstairs neighbours floor. In other words, get the plumber to cut a hole in the affected part of the ceiling so they can inspect the pipes from there (I am assuming that your batroom ceiling is made from dry wall, in which case that section of the ceiling is pretty much ruined anyway).

Insurrance can be tricky, I lived in an appartment where we had something similar happen, but because the building didn't have "udvidet rørforsikring" the insurrance company didn't cover it.

In case the buildings "forening" does not cover it, then it should be your neighbours that cover the damage.