Luke thank you for taking the time to describe your situation.
This is a really difficult topic because there are so many causes of mould and apparent dampness. It is often difficult to pin point the cause of any mould. The most common causes include: lack of ventilation or circulation, hanging washing indoors, inadequate heating and more rarely design failure (classically blocks of concrete flats and worse with Crital or metal windows).
Dampness arising from first floor upwards is usually condensation but could be from a leaking gutter, leaking storage tank or cistern or a missing roof tile.
A bed or wardrobe positioned against a cold wall without cavity insulation could result in mould. The condensate from breathing condenses on the adjacent cold wall is an ideal habitat to attract spores and mould.
This can attack lungs resulting in breathing dificulties.
Dampness restricted to the ground-floor can be caused by a poor damp-proof course or a mound of earth against the outside wall bridging the damp proof course causing really serious dampness with black mould.
A flat roof has a lifespan of say 20 years and should be recovered before break-down allows leaks. The sudden spread of mould you describe would fit this situation - except there may be no flat roof - I do not know!
The recent storms experienced this year has increased such incidences.
I have not heard any bad reports about your letting agent and suggest you give them an opportunity to put this right before taking the matter further.
All this said, under the HHSRS (Housing Health and Safety Rating System) all landlords are responsible for ensuring that property defects do not impair a tenants health. My advice is to advise your agent that if you are not satisfied with their response you will go to the department of environment at the local authority and explain your situation. A visit will be scheduled and the landlord will be advised what if anything should be done to improve your accommodation. Significant fines can be imposed for landlords neglecting their responsibilities. You and your family are the responsibility of your landlord when your health is at stake whilst in their property, which is your home.
This does not answer all your questions on compensation etc - this is merely a starting point to establish the root of the problem and whether you need to take further action. If after the house has been inspected your letting agent fails to respond to any recommendations to improve the problem, let me know and I will attempt to guide you to the next step.
Do note. I am not qualified to advise you and all advice is provided free of charge without responsibility as to the accuracy of the information shared.