seven of the amps have labels above the switches. of those seven, six appear to be hooked up together in a daisy chain config. It sorta looks like the top right amp is only hooked up to the bottom left two amps, one of which sports a 6-band in a frown curve before it's input. So that could be three amps in each of two setups. And that's presumably how the Eruption changeover happens. The Univox is left to oscillate and then Ed is handed a guitar and in one show even hits a pedal on his board before playing the Destroyer.
Here's a question. I'm not big on delays so I've never thought to try anything like this... but does an oscillating delay keep oscillating if you remove the input cable? If so that would provide a few seconds for everything to be patched over by hand. If Ed's guitar lead goes offstage or he's using a wireless (which we see he has at least by the recording of VHII) then the cables can all be swapped by hand in a few seconds while Dave does the trademark "EDWARD VAAAAN HAALEEENNN" yowl after Ed finishes every Erpution

How exactly the changeover happens is still a bit of a mystery... Why has no one tried out a metal panel 1959 for live Ed tone?