The one time, many years ago, I had damp patches on a wall there was a specialist damp company in the phone book I got to come out and check it. Couldn't tell you costs to fix though.
I would suggest getting a couple of builders out to give an Estimate -they should do this free of charge - The stonework around your window may need re-pointing especially if its sandstone. Its difficult to estimate the cost, but work on the assumption of £15-£20 per hour so i would reckon minimum £150 to account for one days work.
Stone walls are cold B-have. It's condensation I expect.
I think you need an electrician to install an extractor fan in the ceiling (over the shower if you have one).
The fan can be ducted through the roof or connected to an outlet in the soffit ( that's the horizontal bit between the fascia board and the wall ......... just below the guttering)
Should be around £150 depending on how tricky it is.
The spoiled paintwork should then be primed with a stain-blocking sealer, and just re-painted.