Depending on the type of propeller, one may or may not see a change as oil pressure is lost. With most conventional Hartzell and McCaulley propellers on light piston airplanes, the propeller will revert to low or course pitch (high RPM position). This is also generally the case with a propeller governor failure. One sees the propeller speeding up. One must keep throttling back to keep the engine RPM under control, because in essence, the propeller has just become as a fixed pitch prop. It's held in the high-rpm (low pitch or course pitch position) by aerodynamic twisting force (ATF), spring pressure, and/or nitrogen pressure in the propeller dome.
Some propellers need oil pressure to feather, and some don't. The Hamilton Standard Hydromatic propeller needs oil pressure to drive the propeller into feather. Hartzell props don't, and will feather with a complete loss of oil pressure. Some will feather automatically, some will not, depending on the control system in use. In turboprop installations, some will feather, some won't.
I experienced a complete oil pressure loss last year in a piston installation, and the propeller was featherable afterward. I experienced a complete oil loss in a TPE-731 turbine installation in a single several years ago, and the engine ran as though there was nothing wrong (honeywell says it will run a half-hour in that condition)...but had no torque because I had no control over the propeller...which did not feather.
If you lost engine oil pressure in a 172 with a constant speed propeller, the propeller would not go into low pitch right away. The mechanism for propeller pitch changes is a lot like a car jack, supply pressure to lift, trap pressure to hold, let pressure out to drop. So initially when pressure was lost, and before the engine started making serious metal, oil trapped in the hub would hold the pitch in place, assuming you were just flying straight & level, making no changes to demands of power or propeller pitch.
Generally, that is not the case, and I have found the opposite to be true in actual practice.
With any demand on the governor to alter the pitch of the propeller (which occurs constantly in flight), the oil pressure will be released but not resupplied, and a change in RPM will occur. The governor, flyweights, and propeller pitch doesn't stay constant inflight; these vary position to maintain a constant RPM. This works, until oil pressure is no longer available to continuously supply the propeller via the governor.
Again, what happens in a given airplane and propeller/engine installation depends on the installation; some will autofeather, some will revert to high pitch, some will revert to low pitch.