Unless the NDA includes a specific statement to the contrary, its existence would have to be revealed in response to an FOI request. This is one of many reasons why the agreement should be drafted by a specialist solicitor.
There is, of course, no penalty for disclosure under an unenforceable NDA, so it isn't worth the paper it's written on.
A standard local council insurance package includes access to a team of legal experts ready to deal with such matters on your behalf. You are obliged as a condition of your insurance to notify them of any claims against the council, or disputes that might result in future claims, so failure to do so may invalidate your insurance. Once the insurers take over, the council will not be involved in the process, so confidentiality is less of an issue, as you can't disclose something you don't know. It also means that financial settlements are agreed by your insurer as a claim against your policy, so do not appear in your published accounts.