The ECU doesn't control the starter, it knows nothing of it. If it's a standard Caterham setup then there should be 2 wires to the starter, a heavy-duty red cable capable of taking 300+ amps which will be directly to the battery +ve terminal or the master switch if you have one, and a thinner one which comes out of the loom and energises the solenoid.
Does the solenoid click when you turn the key? If not then check for 12v on the thin wire when you turn the key - if it's there then it sounds like the solenoid is duff. If it's not there then either there you've got the wrong wire or there's a fault somewhere else in the wiring (most likely at the ignition switch end). I'm not sure whether there's a relay between the switch and the solenoid. You could try taking 12v direct from the battery to the low power connection on the solenoid (make it's in neutral first...)
Mike