Usually the fuel pumps come on right away to get fuel into the carb before one cranks it over. I'd bypass the switch as well, maybe the hot oil and the bearing clearances result in low oil pressure at idle. If you get it fired up and the pressure doesn't come back shut her down. Hard to believe a bearing would go or the pump would quit just idling for a few min.
Is there a five quart puddle of oil under the car?