The council's insurance will cover them (not you) for two things:
Firstly, their insurance will cover them for any damage to the building (not to your property) caused by such things as a burst pipe. This is because the building is their property and, as a duty to council tax payers, they're obliged to ensure that they keep the building in good condition and avoid large bills for repairing unexpected damage.
Secondly, their insurance will include 'third party cover'. This insures them against a successful claim being brought against them because of their negligence (or the negligence of their employees or agents). For example, if a council employee had carried out a poor-quality repair to the pipework in your loft, resulting in flooding and damage to your property, the insurance would cover any claim by you, for repairs and replacement of your property, because of the negligence of their employee.
However, it seems extremely unlikely that the burst pipe could be attributed to any negligence by the council. Your only chance of proving negligence would be if, for example, you could show that the council had told you that they had lagged the pipes in your loft but they had actually failed to do so.
As stated, the council's insurance only relates to damage to your property caused by their negligence. Any other cover is your responsibility. So, unless you can prove negligence occurred, neither the council nor their insurers should pay for the damage.
Chris