First you should be aware that the gauge in that is essentially an idiot light. It isn't hooked to a transducer but rather a pressure switch (at 5 psi I believe). When the switch closes the gauge reads in the middle.
Assuming you have checked your oil level and it is good, as for what could be going on, In no particular order.
Possibility #1, the oil pressure is actually low until it gets warm. Given the switch closes at fairly low pressure, that is concerning. Don't be revving the engine while cold.
Possibility #2, pressure is coming up but the pressure switch is just sluggish for some reason. Maybe the contact is going bad, maybe the mechanical movement is sticking.
Possibility #3, something is faulty electrically. A bad contact, or maybe even the ground (picked up at the block) isn't good. Some heat and vibration causes it to make the final connection.
Problems of this sort are not uncommon and they most often are not a real oil pressure problem. When the engine is cold, pressure tends to be much higher since the oil is more vicious and causes more back pressure so pressure probably isn't coming up slowly.
You could unplug the switch and use an ohm meter and see how the switch responds at startup. If it is still delayed you know it isn't issue #3.
Replacing the pressure switch may solve the problem. This tends to be the most common and it is cheap. Just make sure when putting it in don't use so much Teflon tape as to prevent good connection to the block. If the problem goes away you know it was the switch.
The only way to really be certain is to plumb in a real pressure gauge. If you want to retain the dash gauge, a tee can be used to allow for both.
We had the same problem on our 99 Ranger with 3.0 a year ago. When looking into it, and having had the engine fail as a result of real oil pressure loss the year before, we were very risk adverse and we had seen some reporting that replacing the switch didn't help. So we used a tee and plumbed in a real gauge in parallel with the switch. That worked great.
We can't say what was up though because after we did that the switch always closed and the dash gauge came up immediately so we could never "catch it in the act" to see what the pressure was when it wasn't indicating. The switch worked correctly after that. Our guess is that either the connector at the switch had been flakey, it had a flakey ground or some dirt had gotten in and blocked the switch port. Whatever it was, working on it cleared the issue. We are happy to have the gauge,
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