Irrespective of whether there's any insurance in place, anyone who suffers loss or damage, through the negligence of another person, is entitled to claim (and receive) compensation from that person. (If there's no negligence or intent involved, then no claim can succeed).
Your tenant is effectively claiming that you were negligent for failing to unblock the bath overflow. However, unless you had previously been made aware of the problem, he almost certainly hasn't got a leg to stand on.
You, on the other hand, are claiming that he was negligent for failing to keep a check on the water level. If the relative merits of your arguments were to be tested in a court of law, I'm sure that you would be the victor.
That means that it is the tenant who is responsible for paying for your neighbour's repairs. If he hasn't got insurance to cover it, he will be obliged to find the money out of his own pocket.
That's a summary of the legal position but, as far as retaining good relations between yourself and the neighbour are concerned, it might not represent the most practical way of dealing with the problem. i.e. if the tenant has no insurance and a zero bank balance, the neighbour still won't get the repairs paid for, even though he has a legal right to receive compensation from the tenant.
It might be worth having a word with your insurers. They might consider a claim under the 'third party' provisions of your insurance. (It's more likely to be the buildings insurance than the contents insurance). However, if they did so, they would be entitled to make a claim (through the courts if necessary) against the tenant, to attempt to get the money from him.
Chris