Painting the exposed chimney with silicone will stop the rain from entering the brick work and cannot do any harm.
However!
The usual senario for a damp chimney is.
Where the back of the chimney is uninsulated, ie; not in the middle of a house. And where the chimney is the coldest spot in the room.
The air inside the chimney starts a circulation transfering the heat from the room across the chimney and out into the garden.
The water vapour that collects in a house is held in the air and because air that is damp is lighter than dry air it rises moving to the coldest spot in the house to condense.
In your case it is this chimney, although I would expect there to be condensation on the bedroom windows. It usually forms when the outside temperature drops below 11C and the curtains are drawn, removing the warm passing air the keeps the windows dry during the day.
This problem is usually down to the heating being intermitent. When you turn the heating off or down, the airs ability to hold damp is deminished.
For example, air at a temperature of 30C can hold 30 ml of water per cubic metre.
When the temperature drops to 20C it can hold only 18ml.
At freezing its almost dry.
The solution, some or all of the following.
Open the window and let the water vapour out along with the heat in the room. (expensive)
Or keep the heating on all the time at a steady temperature.
Or insulate the chimney breast.(the damp will go elsewhere)
Or run a de-humidifier all the time.
Or change the way you live.
Water vapour comes from cooking, washing, drying things on radiators.
Using extractor fans in the kitchen and bathroom while and after cooking and washing, is a great help.
Keeping the kitchen and bathroom doors closed at all times will eliminate most water vapour.
Get rid of flowers and pot plants.
Or wait for the warm dry days of summer when it will go away.(until next winter)
Finally, breathing and sweating, our breath is saturated with water.
We each breath out 2.5L of water vapour per 24 hours, children and animals more, old people less, this goes straight into the air.
You need ventilation in proportion to get rid of this, an upstairs window open for five minites each morning will help.