The burnt wire was in the transformer itself, it's all the visible evidence I can find. I've got a replacement transformer of the same values but it's too big to fit inside the box, I'm having to temp wire it up outside untill I can find the right sized one.

The only fuses (other than the 13amp plug) are the MCB's, none blew or tripped.

I've got a multi-meter but don't really know how to use it, I'm OK at ID'ing what voltage from what wire & continuity etc but please don't ask me to start measuring OHMs or amps etc.