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Same as other resources for the country overview map, you need to advance turns after constructing the building which produces the resource.

That's where I'm stuck. To build the building I need more furniture but the carpenter is out.

I don't understand what you mean with "the carpenter is out" then. Did you upgrade the work area after the construction, to get the carpenter? 
The npc when you enter the work area via the worldmap is not of importance, only whether you upgrade the work area or not.