Graham, a couple of things come to mind, but the easiest way to isolate the problem is to use a simple, ground defeater plug, which would isolate the ground systems from each other. They're available in any hardware store, Home Depot, Lowes, etc...

Another problem could be your venue ground system, which is not unusual in older buildings, mainly churches and theaters. When I worked for Executone Sound Systems we frequently had to install earth ground systems in order for their sound systems to operate without encountering 60-cycle/HZ hum. The big tip off there was when people complained of getting bit by their microphone - an electrical shock when their lip touched the mic frame and they were standing on a bare, damp concrete floor. ZAP! This can be tested with a multi-meter, but if you don't have electrical circuitry expertise, this is best left to an electrician.
I've plugged just about every electronic musical instrument into my Bose L1 Compact that can be, and only once encountered a hum - and it was from a video recorder that had an external power pack (wall wart) plugged in. It hummed like crazy until I plugged in a ground defeater and the problem immediately ceased. I wish I lived a bit closer Graham, then I could come over and solve the problem in person. But you are about 900 miles away.
Chas, Another Green Coconut Margaretta sounds like a splendid idea right about now.
Hope this helps,
Gary.