Without being mellodramatic or alarmist about it, a dust mask would be sufficient to protect you from the spores of a mouldy carpet – it is unlikely to contain anthrax!! However, as spores can trigger allergic reactions, a dust mask would be prudent. You’ll pick one up at a DIY store. Bear in mind that some of the agents used to treat mould, benzalkonium chloride or hydrogen peroxide, can be quite toxic. So have the extent assessed by a professional surveyor before going any further and spending money getting in a specialist. If you feel strongly enough, have the carpets taken up and disposed of. However, in order to have some chance of recovering costs from either the builder or your insurance company, keep records (photos, letters) of what you are doing – don’t just incur costs without first of all giving the other party time to rectify their wrong doing.
The 'health board people' (who are they?) probably wouldn't be too interested in this, it is the job of the local authority to enforce Building Control, so it may be worth while having a chat with them. Did you have asurveyor look at the property before you bought it - speak to him / her if so and see if they'll visit to assess the situation. Without the exact parciculars of your apartment, it is difficult to pin point the problem on a chat forum. It is unlikely ‘the health board people’ would also not do you a report. It is also unlikely 'the health board people' will sent a SWAT team to analyse your house for a bit of mould. You have done the right thing by contacting the management company, also consider pursuing the builder directly as it is him you had to build the apartment and you had the contract with, not the management company. See how they respond - I would take pictures of the damage and keep records of conversations, letters for future reference (possibly from an insurance claim or solicitor to refer to).
Before taking the drastic measure of moving out and incurring costs and the inconvenience that you may not be able to recover, you should seek legal advice if the problem is not rectified to your satisfaction. Negligence will firstly have to be assessed (i.e. building regulation compliance, compliance with construction best practice) on the builders part before you incur any significant legal costs.
On a positive note, I had a friend move into an apartment in 2004, the same problem occurred. Treatment with the de-hudidifier did the trick, although it did take several weeks to de-water the place entirely. You might be able to recover the cost of this from the builder.