It would be usual for the clerk email address to be a generic one and for the new clerk to be given the password immediately. If the old clerk needs access (not sure why that would be) up until he/she leaves, then they can both access with the new clerk changing the password when the old clerk leaves. It is not acceptable for the old clerk to retain access to council documents/emails after the date he/she leaves.
Also usual practice for the passwords used by the clerk to be written down in a sealed envelope which can be opened by other staff (if there are any) if/when the clerk is no longer able to act suddenly and where there are no staff, that envelope is usually handed to the chair on the understanding that it isn't opened unless absolutely necessary. Passwords should be changed regularly of course.
Difficulties can sometimes arise when an account uses two stage authentication such as a text message to confirm identity. If the clerk is using a council mobile, it's not a problem (assuming the mobile is passed on, as it should be) but can be problematic if the clerk's personal mobile is used. In this day and age, even the smallest of councils should be providing their clerks with appropriate equipment (laptop etc.) and a council mobile phone.