One of two Nest Thermostats won't run Fan Only, fan runs with heat though
To troubleshoot this, you can remove the Nest and use a jumper wire to touch the red and green wires together.  The fan should come on.  If the fan does not come on, then the system is not wired correctly on the other end.

You'll have to take a look at the air handler side of the wires to see what happens to the green wire.  If the wire was broken or if they were trying to get around a limitation of some other thermostat, the could have jumped the fan wire to the heat or cooling to get it to work at the expense of not being able to run with the fan only.

The fact that the white wire doesn't go to the W terminal on the Nest already proves that something is out of the ordinary.  It might not be broken, but there might be a combination of equipment that requires some odd workaround.  That or the person that tried to get it to work last time didn't know as much as they should have.