The NALC publishes a model disciplinary policy that complies with the latest ACAS model and guidance.
In relation to discussion at a full council meeting, the policy states that "Information about an employee’s disciplinary matter will be restricted to those involved in the disciplinary process. A record of the reason for disciplinary action and the action taken by the Council is confidential to the employee. The employee’s disciplinary records will be held by the Council in accordance with the General Data Protection Regulation (GDPR)."
You mention that the Clerk said that he would give the employee a verbal warning. Did the Clerk make this decision himself? We are employed to act upon the instructions of the Council and have no powers to make decisions on their behalf. If the Clerk refused to follow a direct instruction from the Chairman of the meeting, the Clerk should, at the very least, be reminded of his responsibilities and the standing orders relating to the conduct of meetings.
Under the model policy, a single Councillor who has had no prior knowledge or involvement in the allegations is appointed to investigate the issue. The investigator then reports his or her findings to a sub-committee (assuming you have some form of staffing committee) of three other Councillors, none of whom has prior knowledge or involvement in the allegations. The role of investigator may be undertaken by somebody from outside the Council, but only where this is necessary in the absence of independent Councillors able to undertake this role.
At the start of the process, the employee must be given written notice of the allegations and a copy of the Council's disciplinary procedure. They must be given the opportunity to invite a colleague or union representative to any meetings that take place to discuss the allegations. Written records of all discussions must be kept and provided to the employee.
As you suspected, the way this has been handled thus far is a complete mess and it may no longer be possible to turn this into a valid disciplinary procedure. My advice would be to draw a line under this matter and to ascertain whether the employee would be willing to attend a mediation meeting with the two councillors who have had no prior involvement, with the option of bringing a colleague or representative, in order to ascertain why the task was not carried out and clarify whether this can now be done. For mediation, the tone of the meeting will need to be supportive and aimed at establishing the facts, without any disciplinary tone.
We can't change the past, but we can the future.