Rich is right on: swapping plugs on the relays, then checking for fuel pump operation is the easiest. My relays were originally on the firewall, too, but I moved them both, when I moved the battery. They joined the fan relay, as neighbors.
I bought a ($12; Chinese made) new relay from Chevy, which now sits unused, as I just solved a problem like yours. Mine turned out to be the 3 & 4 gear linkage switch. Turning the switch 'on' (+12vdc) sends a 'request' for O/D to the ECM and turns on the dash light, but the ECM won't allow O/D until closed loop ops. This is not, strictly speaking, triggered by a temperature switch.
The dash light coming on means the +12vdc side is okay, at least to the linkage switch(es). The pressure switch is inside the Nash, and not directly testable w/o dropping the pan. The external lug is merely a pass-thru, and not of too much value when troubleshooting, except that the +12vdc can be found, if the switches and relay are okay and operating, i.e. sending the voltage there.
The other half of this circuit, the GND side is controlled by the ECM (as above), again, though the relay.
If the late O/D 'on' is occurring in a consistant manner, it may not be a problem with the relay's contacts, which can be 'dressed' up, with emery cloth and contact cleaner, after the cover is removed.
Please check for proper fluid level in the Nash, especially if the unit cycles, particularly under lateral/longitudinal G's. Years ago, before any serious mods, I fixed a cyclic O/D by changing spark plug wires. Honest, I did nothing else! While the unit used to 'come on' exactly at 174* temperature, that trigger temperature had raised significantly; then became cyclic, even at steady, freeway speeds. The wire change restored the original operation.
FWIW,
I believe that my mods and perhaps weak wires had caused a rich condition, which was sensed by the oxygen sensor, thereby, the ECM said "Open Loop" >no O/D. My tailpipes were blackish, too.
There have also been reports here, of problems with the console switch. It switches the 12vdc that it is fed.