Overheating when idling, radiator fan not starting

Hi
I slowed down to a crawl in motorway traffic and I had a warning light to say the engine was overheating. Pulled over to let the engine cool down and turned round to drive carefully home. I've only had the issue since by running idle for about 30 mins as the engine slowly heats up.

I understand that the radiator fan should start spinning once the coolant has reached a certain temperature but it doesn't. Nor does it spin when the A/C is turned on.

I've bypassed the fan control module and run the 12v straight into the fan and it spun up so I've determined that nothing is wrong with the 12v fan fuse or fan itself. I got a replacement control module from a breakers yard but it doesn't make a difference. Granted the replacement could be broken also.

Now I'm looking at the smaller diameter cable which ultimately joins the 12v lines going into the control module from the thermometer and other triggers. I'm guessing that if there is an issue with the thermometer that turning on the A/C would still start the fan at a slow speed? Neither start the fan so could it be a relay I've read about? I'm struggling to find the relay, doesn't appear to be in the fuse box close to the battery.