As the others have said, wash the ceiling. I would use dilute bleach.
Again, as they suggest, you have airflow problems, It also sounds like you have matt emulsion on the ceiling, ideally a ceiling subject to high levels of moisture would have water based eggshell.
Over the years I have painted ceilings in bathrooms with inadequate ventilation. The moisture forms on the ceiling and literally drops down to the floor, but it doesn't develop black mould, and it were to, it could be wiped off with ease.
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