No - public servants are not eligible although a lot will be working from home.
I thought that furlough was only available to organisations up to a certain size. A lot of the work that takes place in the Civic Centre building relates to meetings of a kind that can be relocated to online video meetings etc, and this must reduce the need for use of Council parking permits etc.
I don't work for the council but have just finished my second two year term as a non councillor Scrutiny representative, and those meetings where they have taken place have all been online since March, and this will mean that officers, councillors and anyone else involved is probably logging in at home and won't be having to drive home from a meeting which ends at 10 pm or later! Also reduced need for Reception and Security and working out of normal office hours presumably for those staff.