Jonfis, even in the UK you would get the conflicting advice as water ingress is often difficult to locate. However, I would start with a dehumidifier to get rid of the airborne damp and that which you create (Showers, cooking, kettles etc). That will/should get rid of the black mould/spores higher up the walls. That will also help any other damp.
Do you have vents on the bottom of the property outside? Make sure they are not blocked. If you vent under the property there is less to track up the walls.
With regard to the rain from above, wait till you are a bit drier from the dehumidifier and re visit it. You could after all be treating damp from inside living. You could spend a fortune to find you haven't cured it and have to start again. Take it one step at a time and to be fair generally I would have thought due to the climate the ingress from above would dry out before it gets in unless it is a significant large crack/entry point which would be easy to identify.
Buy a good dehumidifier and you may eradicate or reduce the issue to a point where you can see the issue.