TCP and Medicinal flavours are almost certainly caused by chlorophenols (chlorine+phenols). Treat your water with 1 crushed tablet per 10 gallons of cold tap water before use.
As for the concentrations or taste thresholds this will probably vary from person to person. If you have chlorine or chloramines in your water though, there will be chlorophenols present in your beer to a greater or lesser extent.
The reaction after adding the campden tablet is very quick, I don't have any information on exactly how quick but it is a matter of minutes, probably less. If you add it then pour yourself a beer it should be fine.