If you want anything close to an accurate answer. You're going to have to tell us a bit more about the wiring between the thermostat and the furnace, or include some photos of the wiring. And include make, and model numbers for all the equipment involved.
Most furnaces manage the blower themselves during a heat call, i.e. the thermostat simply energizes the W wire. Some also manage it during a cool call, but not all do. So when calling for cooling, the thermostat usually energizes both Y for cooling, and G to turn on the blower.
Since you say the blower doesn't come on for cool calls, or when the fan switch is moved to the ON position. That points to a problem with the "fan circuit". Between the thermostat and the furnace, there should be a wire (usually green) connected to the G terminals of both devices. When this wire is energized, it tells the furnace to start the blower. Some furnaces have an on/off delay, so the blower may not come on immediately. On some units, this delay is adjustable.
Since you haven't told us the make and model of any of the equipment, there's no way to be specific about what happens in your furnace. Some furnace control boards have terminals for HEAT, COOL, and FAN. While others may only have HEAT, and/or FAN. The motor speed wires are connected to these terminals, to allow the furnace to run the blower in different speeds depending on the situation.
The first thing I would do, is test for voltage between the grounded ("neutral") and the COOL terminal on the board, during a cool call. If you measure about 120 volts (line voltage), that means the board is energizing the terminal to turn on the blower. This means there's a problem between the board and the motor, or with the motor itself. If the terminal is not energized, then you'll have to start moving "back" through the circuit.
If the furnace is signalling for the blower to turn on, but the blower is not starting. You could try connecting a different speed wire to the terminal. Since the blower works for heat, you might try putting the wire that's on the HEAT terminal on the COOL terminal. If the blower fires up, you know it's a problem with the motor. WARNING: Don't run the system this way for long, you're just checking to see if the blower starts.
If the furnace is not signalling for the blower to turn on (no power to COOL terminal), then you'll want to start by making sure the thermostat call is reaching the furnace. Disconnect the wire connected to the Y terminal, and set the thermostat so it's calling for cool. Measure voltage between C (or the common side of the transformer), and the wire that was connected to Y. You should get about 24 volts AC. If you don't there's a problem with the thermostat, or the wire between the thermostat and furnace. If you do, then the problem likely lies within the control board.
Without more details about the system, it's impossible to offer any more specific advice. It's possible that there's an open limit, which is preventing the blower from turning on. However, without a schematic, there's no way to know if there are any limits that might be involved.
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Thank you for pointing me to the problem - we were too close to see it. Replacing the thermostat cable has resolved the issue - it was pinched between ducts in the attic and had worn through the insulation and developed a short circuit on the green wire.