Can I ask is your cistern porcelain ? in the cold weather mains water is coming into your house at about 5 Celsius and storing not much more in the cistern tank which will sweat like mad in room temp. This could be the start of your cycle of continuous wetness?
Cheers , Steve

Is there a shower installed in there?

First priority is making sure its not from a sinister source like rising damp, leaks from supply/heating/waste pipes or damp from the outside brick.

There has to be a way for the moist air to be able to be expelled from the room. Is there a window fitted? as trickle vents can be fitted over a series of 10mm or so holes drilled in a line through the frame above the window from inside to outside and then 'sticking' on the vent to cover the holes. Inexpensive and a good help in combating this. £5-10 on Ebay for the vents and a careful fairly competent D.I.Y.'er to do.

Ideally would recommend re-drilling a new/going through the old air brick with 4 inch hole and fitting a plastic grill outside and a neat plastic or aluminium cover inside to provide proper ventilation.

Fans are ok as an addition and are a requirement for bathroom overhauls and new houses but can be a pain to install in certain settings without disturbing surrounding surfaces and finishes and cannot perform miracles... probably nearer £100 in total for this option depending massively on the location/access etc.

Any circulation will help. leaving the door open will make a difference and/or small circular vents can be drilled into doors also helping. Very inexpensive - B&Q job.

Heating from radiator or towel rail will also help.

Getting a bathroom installer that will be familiar with handling moisture and damp etc. will be able to offer advice when he/she sees it.

Hope this helps!

Justin