Since oil pumps are positive displacement, the pressure developed by the pump should almost instantaneously increase when the engine starts if all of the oil passages are full of 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. Even if cold oil has not gelled and is pumpable, it may still be viscous enough for the pump to draw oil out of the sump faster than it can return to the sump, thereby causing a vortex to form when the oil level drops low enough.
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.
Go easy on the gas pedal until the engine oil has warmed up!