Just to clarify, a 60Hz synchronous motor turns at 2400 RPM and if it's used in a mechanical timer or clock it's usually with a gear train in a sealed unit with the output shaft turning at 1 RPM. As such the likelihood of one producing RF hash is nil, I wouldn't discount any cam switches it may be driving however. The possibilities are endless and diagnosis by remote control impossible, you could be bantering this about forever. The only solution is good old fashioned hands on detective work, process of elimination. Get out of the bathroom, that's NOT what I meant!
Just remember that many devices stay on when switched off so the way to be sure it's completely powered down is to pull the plug and wait at least a minute for a big electrolytic to discharge. That's by design so momentary power glitches don't upset the apple cart. That made me think, I have a VCR that keeps time for forever and a day, the internal clock must have one hell of a big cap in it.
"A conundrum, indeed, I shall ponder it further."
A conundrum, indeed, I shall grok it further. There, all fixed. Grok means to understand so thoroughly that the observer becomes a part of the observed, as part of the offending device you will gain knowledge as to how to fix it. Then if all else fails you can eat it.