A few consolidating comments.
I agree there is in all probability a block policy that will cover the fabric of the building for water damage either from the external pipes or the leak in your bathroom. As the policy is effectively in joint names of all the leaseholders, the question of who may be 'responsible' for the leak is immaterial as the funding of the policy is shared by all, the insurer pays for the damage and nothing further happens.
The second question is whether the flat below has suffered damage which is not covered under the buildings policy (i.e. to its contents), or if for example there is an excess which the occupier of the damaged flat has to pay and tries to claim from you. In order to pursue a claim against you, your neighbour would indeed have to prove negligence on your part - on the basis of your description, you have not been negligent as you have done everything reasonable once the matter was brought to your attention. Therefore your (Contents) insurers (Contents insurance will have a liability extension) will fight any claim that the neighbour tries to make against you. If he sues you, they will instruct solicitors on your behalf to defend his claim. If he wins, they will pay the award and costs on your behalf. If on the other hand they believe you have been negligent (although again on your description I can't see how they would come to this conclusion) they will agree compensation with him without the need to go through the courts.
Either way you will be protected.