If you know this stuff I apologise, but just in case any of this is helpful I'll type it anyway:
- test for 16V AC across input lines from the transformer
- check PCB board for output DC voltages (+12, -12, +5, and green might be -5? or maybe the return for the power switch? Not sure)
- if PCB outputs 0, retest with motherboard connected (some supplies need a load to turn on)
- if still outputting 0, disconnect PSU, and check the resistance between the coloured wires and ground (if any show near 0 ohms, that'd cause the PSU to shut down generally - assuming it has cut offs) **
- if power output is good, but machine does nothing, Supersoft ROM time
If the transformer is what's broken, I'd also check the power switch in it is actually still functional.
** unlikely as I don't see many tantalum caps which are the usual cause
External power brick inside the case cracks me up
Motherboard somewhat remind me of the Commodore PC-10/20 series.