Engines require oil FLOW and oil PRESSURE is related to flow and viscosity. Oil pumps are positive displacement, which means their flow is proportional to speed. The pressure measured by an oil pressure gauge is typically at the oil pump or in the main oil gallery.  Once the oil reaches its destination (tappets, bearings, etc), its pressure is essentially atmospheric (ie 0 psi) or close to it so the measured oil pressure is what the oil pump develops in response to the flow it is generating.  Excessive pressure can cause oil filters to rupture or oil pump gears to shear so hydraulic pumping systems always have a safety relief valve to limit the pressure developed at the pump.  Excess pressure is bled off by the relief valve back to the sump so you're better off keeping the oil's viscosity to the lowest that protects the bearings.

As you can see by the following photo, engine oils become more viscous (thicker) as they get colder but the 0W-30 oils flow far better than 15W-40.

 

Engine Oil Viscosity Comparison

 

Since oil pumps are positive displacement, the pressure developed by the pump should almost instantaneously increase if all of oil passages are filled with oil.  When the ambient temperature starts to get low, you've probably noticed that it takes longer for your oil pressure to come up or for your oil pressure warning lamp to go out. The reason for this delay is that excessively thick oil is causing one of two things: vortexing or cavitation.

Vortexing results from the oil becoming so viscous that it "gels" and becomes a semi-solid material that cannot flow past the oil pickup screen and into the oil pump.  Instead, the vacuum created by the oil pump allows air to pulled into the oil pickup, thereby causing air-binding.  An air-bound pump does not flow oil and therefore is unable to build oil pressure.  See Thermal History of the Engine Oil and Its Effects on Low-Temperature Pumpability and Gelation Formation.

Cavitation is similar in that that oil is too viscous to flow through the pickup screen as fast as the oil pump requires. The vacuum in the oil pickup causes the oil to "flash" into vapour bubbles, which then collapse in the oil pump.  A cavitating pump is inefficient and is unable to generate as much flow as it should.  Severe cases of flow restriction can lead to flow-limited failure.

Richard Widman makes this observation on page 2 about hydrodynamic lubrication:  Selection of the Right Motor Oil for the Corvair and other Engines:

A cushion of liquid oil surrounds the lubricated item and holds it away from the rest of  the parts.  When the proper oil viscosity is used in a properly built engine at operating velocities, the crankshaft is in hydrodynamic lubrication.  It has no contact with the bearings.  The only physical contact is during startup before velocity is attained or under lugging from improper gear range.  If the oil is too thin, it can be displaced and allow contact. If it is too thick it takes longer to get to the bearings and valve train as well as build pressure (the cushion) in the bearings creating additional wear.

To ensure that engine oil flows well to the oil pump at low temperatures, the oil must have adequate pumpability.  The "W" (w means "winter") viscosity rating (0W, 5W, 10W, 15W, 20W, 25W) of the SAE Viscosity Grade is what describes how pumpable the oil is at low temperatures.  Pour Point Depressant additives are sometimes added to prevent gelation by reducing low temperature wax crystal formation.

Depending upon the design of the engine and the orientation of the oil filter, another thing to consider is leakage from the anti-drainback valve (ADBV, check valve, non-return valve) in the filter.  If this valve does not seal properly, oil will drain out of the filter and, no matter the ambient temperature, there will always be a delay for the oil pressure to come up.  If this is the case, you need to find another filter with a better ADBV.