You certainly need more ventilation to let the warm wet air out.
The cistern will be the coldest part of the room. Have the window open for at least part of the day. Drape a towel over the cistern as an insulation blanket. Keep the door closed to keep steam and warm air entering. Turn down the "wick" on the radiator.
Condensation is to be expected if warm air comes into contact with a cold object. Buy or borrow a dehumidifier and run it in there for a few days.
If it's adjacent to your bedroom and you leave the adjoining door open then that won't help - your natural night time breathing will just add to the problem so try to have a small window slightly open in the bedroom if you can stand it.
The cistern has cold water entering it.
Warm wet air condenses out on the cold cistern
You need to control the warm wet air - you cause it.
Good luck...
Peccavi
March 2013