Nest thermostat not turning on heat
Broken Nest

Either the output driver for the W output on your Nest is toast, or the Nest's solid-state output drivers can't put out enough current to drive the load of your system.  Try switching the wire connecting the Nest's W output to the Nest's G output and seeing if turning the fan on causes the boiler to fire up.  If it doesn't, then you can't use the Nest with your setup, sorry!