How fast the oil pressure rises is dependent on the oil temperature. If the engine has just been running then the oil pressure should register almost immediately after the engine starts. However the hot oil will be thin so you will likely not see the oil pressure in the green. As long as it is above the lower redline at idle you are OK.
If the oil is cold there will be a delay before the oil pressure starts to rise, but the idle oil pressure will be higher as the cold oil is thicker.
Personally if I do not see oil pressure in 15 seconds on your typical small Lycoming or Continental engine, I would shut down.