Strange coating in old lime mine shaft
Given your description of the base rock and the age of the mine I would suggest that what you are seeing is what in a cave we would refer to as flow stone (A deposit of Calcium Carbonate in one or more of many crystal structures).

Simplified Process:

Ground water often has varying degrees of dissolved carbon dioxide in it.  This carbon dioxide within the water creates a mild acid, carbonic acid.  The acid in the ground water will corrode out Calcium Carbonate in the base rock.  Most ground water will be saturated with Calcium Carbonate, the lime, depending on the concentration of dissolved carbon dioxide in the water.  When the water seeps out of the walls of the mine it has an opportunity to off gas some of the dissolved carbon dioxide.  When this occurs the water can no longer hold as much dissolved Calcium Carbonate and the excess is deposited on the walls in various forms.  The mechanism of dissolving the rock and re-deposition is driven by various equilibrium reactions( involving temperature, water mixing, partial pressures of the gases...).  A karst hydrologist/geologist would be able to give you much more detailed information on the processes. It is this process that forms caves and causes the formations with the caves to form.

The following web site gives a pretty straight forward explanation of what I believe you are seeing. https://en.wikipedia.org/wiki/Karst