From what I have observed just switching back and forth between the two while driving, the oil temp tends to stay several degrees cooler than the coolant temp when in normal operation. But during the occasional bouts of higher RPM loads, the oil temp will rise to a several degrees above what the coolant tops out at (and then falls) and stays at that temp for a little while. And as long as the engine load goes back to normal, the oil temp eventually falls back down below the coolant temp and settles into its normal routine.
Given that, I would not expect the oil temp to rise up the same level as the engine coolant temp in an overheat situation if the alarms and bells and whistles that go off get the driver’s attention and they pull over and shut it down. Now if the driver doesn’t immediately do something g to stop the overheat, I would expect the oil temp to soon meet or exceed the overheated coolant temp and stay there until something happens to stop the overheat.
Either way, I feel like your engine’s chances for long-term health after that situation are much better if
there is some good synthetic oil in the sump. Especially if the oil isn’t changed right away but allowed to continue for the rest of the OCI.