I agree with this. My parents' old house was mouldy. Mostly through a dodgy non-existent damp proof course on the house but the lack of decent heating didn't help either (they also had storage heaters). It was noticeable that the rooms with better heating (storage heater AND log burner) had less mould.
As a quick test, you can buy a cheap thermo hygrometer that tells you the humidity of your home. The aim is that it should be around 40%.
Given the levels of mould OP says, I bet theirs is at least 70%. The lack of heating will contribute to this, but if there's mould on the outside too there is probably a problem with water penetration to the walls too. That is definitely the landlord's responsibility, if so.
We use a dehumidifier to dry our washing and it will take the room from 70% with washing in it, down to 40% throughout the day. It dries the washing quicker, gives me water I can use to water my plants with AND heats up the room it's in at the same time.
They're not cheap to buy, but pretty cheap to run, but it will save your clothing and furniture from any further mould.
If once you've done everything you can to keep the property aired and heated you're still getting mould and the landlord isn't doing anything, contact Environmental Health.