I've had all what your experiencing apart from the baby with it which must make it more horrible, it was 20 odd years ago but still remember what a nightmare it was. That was when I used to use storage heaters strangely enough.
I don't know what sort of budget you've got to work from so sorry if this is useless info because I myself know money for one offs such as these doesn't just magically appear because they would help. On the plus side, they are things you can take to a new property if you can find the cash.
You need an electric dehumidifier, can't stress this enough. There's one on offer at Argos at the moment for £100, 12 litre Challenge one. It's the one I've got (only I paid full price) and it sucks the water out of the air really quite well.
You are going to have to strip the wallpaper or paint right back to the plaster, there's no other way to kill the mould as it's spores are right stuck in that and will keep growing back. Then you need to get some Dettol Mould and Mildew killer and follow the instructions.
DO NOT use bleach. I got this info from a bloke at the council who came out to view my mould problem. He said that although bleach will kill what you can see, it won't touch the bits you can't see. Not only that but using bleach leaves behind deposits that attract moisture back to the area and encourage the black mould to grow back even faster.
That all has to be left to dry out. Don't try to repaint it while the ceiling is still leaking, it's a waste of time. Better to have bare plaster, easier to dry out.
Contact environmental health instead of the repairs management department, they couldn't give a shit how much your suffering. I had to do this myself years ago and that was when it got fixed because environmental health made them do it.
Finally, you need to get a electric radiant bar fire such as the Complex Yeominster (£70ish) or similar. If you look up the Dimplex one on Amazon the other cheaper ones will be on the also viewed/related bit.
Storage heaters are absolute shit and the most expensive way to heat going. A radiant bar fire is expensive to run but I personally find it cheaper than the storage heaters I have (I'm all electric, no gas). The heat is instant and because it's radiated heat, you can feel it in your body better and it warms the walls and furniture too so that heat is released back in to the room from them. I find one bar gives more heat than any storage heater. Halogen heaters are not bad because they also put out radiant heat but are a bit bright, burn your eyeballs if you can see it out the corner of your eye. Stay away from fan heaters, very very expensive to run and only warm the air temporaily, freezing again the second you switch them off. Human beings can't absorb heat from the air, we can feel it on our skin which gives a mere illusion of warmth but it doesn't warm our core, we need radiant heat like that from the sun.
I know my shit when it comes to electric heating. I've never had gas as an adult and have done much research and tried it all.