Checking radiator cooling fan

I'm still chasing this overheating issue so further suggestions will be appreciated.
Reminder: MG TF 2002 135 with air. Getting near max temp (almost to the 4/4 point) in stop-start traffic, returns to normal temp quickly when moving. The engine bay fan activates when the car is hot therefor temperature sensor seems to be working.
So, as suggested, the engine fan has been checked and found to work fine when hit with 12 volts, so can now assume it is not getting a signal to turn on as coolant level rises. 
I've now found that the AC isn't working although the refrigerant level is fine and the AC guy suggests it is unlikely that the compressor is faulty, thus suspects it's not getting the appropriate signal to turn on. (The idea was to turn on the air to activate the AC fan thus cooling the radiator, at least until a solution to possible ECU fault can be found.)
Also, I have learnt that the relay sitting between the fuse box and the 70 amp fuse under the front bonnet operates the cooling fans. My question now is, does this relay also operate the AC switching? If so, could it be the relay at fault and not the ECU? Guess I'm looking/hoping for a cause other than the ECU - I've been in communication with Bitz and they don't have the appropriate ECU set as the air-conditioned models there in the UK are rare.

One relay operates only one fan. However since you have AC, at place you mentioned you do not have one yellow relay but black box of 4 relays pack together as you can see in this thread:  https://www.the-t-bar.com/forum/roverlike-s-guides/97846-mg-f-tf-fuse-relay-listing  part number YWB100801. In that box you have both relays which operates cooling fan and AC fan.

Brilliant, thanks Roverlike. 
The info you provided includes Additional Relays and the relay I was wondering about is an 'Air Conditioning Pack Relay Module" which controls the radiator fans + the AC compressor clutch, the items which have failed. 
I imagine the relay will be easier to find either new or second hand and will certainly be easier to fit and less worrisome than the ECU sets.
I'm very grateful. 

If that relay pack has failed, and one can’t be found to replace it. It might be possible to install separate stand alone relays to replace the pack, it would involve studying the wiring of the pack and replicating it to serve the stand alone relays so as everything works as it should. You would only go to this trouble if you couldn’t find a replacement relays pack.