Why won't the fan run independently of the air conditioner?
I would first try touching the R and G together at the thermostat board (the wires coming out of the wall) and seeing if the fan turns on.  If it does, then all the wiring is correct.  If it doesn't, then I would try to do the same thing with two short wires that have been attached to the thermostat control board R and G terminals.  If it still doesn't, then the problem is in the furnace.  If it does, then you have a loss of continuity somewhere between your thermostat and your furnace on the G wire (because the heat and A/C work, so the R works).  I don't have a Nest myself, but I know that not all thermostat systems make it obvious how to run the fan without cooling or heating.