Hi Col.. I'm not sure about the insurance thing - my sister's upstairs neighbour had a leak that came through to her ceiling and she had to claim off her own insurance That said though - it's her flat - this is the HA's so I'd be tempted to check first re the cost of the bills.,. and if you say you're not going to pay to dry the property out, they'll surely do something.
Most importantly though - get yourself to the doc's tomorrow to let him know what's happened and what the various implications might be for you. Is it possible the HA might be able to arrange temporary emergency accommodation for you? If so, get your doctor onside asap. Hope you get it sorted, sweetheart.