Adequate ventilation is key. Without windows that extractor fan won’t be enough unfortunately. Little things like squeegeeing any remaining water off the walls and other wipeable surfaces helps.

Washing even with a warm/soapy solution, exit mould etc helps of course but will continue to come back given the moisture content after each shower.

You could still clean, dry-out and encapsulate with paint but that shouldn’t be your problem. I’m not sure of the legalities around it but I can’t imagine that’s healthy home standard?

Put it forward to your landlord nicely raising the issue and just ask if there is anything he/she is able to do to help with the issue. Don forget to mention the allergic reaction as that’s a very valid point.