OK. Ventlation of the bathroom is especially important to remove the water vapour. Provided the bathroom door is kept shut it will draw dry air in through the gap under the door, and steam will not diffuse into the bedroom. If you can identify the fan used I can see how powerful it is.
Modern extractors with ball-bearing motors can be amazingly quiet, especially if they are mounted on a padded board on the loft, so you will not hear them through a solid door. For auditory privacy I recommend a solid core or fire door on an en-suite bathroom. Not a hollow one.
We don't know if you have a hot water cylinder. This is relevant because a towel rail or other heater can be plumbed to come on whenever the cylinder is heated, typically during and after a bath or shower. If you put a towel rail or radiator in the wardrobe, have a TRV on it. Don't dry damp clothes or towels in there.
The tubular heater mentioned is very suitable for keeping a wardrobe or other small area dry. I use one as an anti-frost measure under some pipes in the garage. The one illustrated is listed as having its own thermostat, but due to the low power requirement, pretty well any wall-mounted CH thermostat can be used. A thermostat means it will only run when cold, making it economical. I have solar panels, so it costs nothing in hours of daylight. They are supposed to get no hotter than a teapot, but I'd suggest getting the wire mesh protective cage, as you might not notice if a garment fell on it at the back of the wardrobe. There are various sizes, the longer ones give more heat.
You can also line the wall inside the wardrobe with insulating foam board, and this can be painted or wallpapered to match. Do not have gaps in the insulation as this is where condensation will form.