The way you describe it, sounds like the pressure switch is working eventually but not reacting right away.
Probably the switch has gone bad and pump needs replacing, but I'm wondering if some sort of clog could cause this... something on the output side of the pump like a bad check valve that only slowly lets water pass into the system after you run your faucets so it takes some time to detect the pressure drop. Or maybe a valve on the output side that isn't fully open?
One thing you might try (as you'll have to do this anyways if you replace the pump) is to disconnect the pump from the plumbing on the output side and see what happens... if the pump is ok then it should detect the drop in pressure and immediately turn on and you know to look for something downstream. If it doesn't then you know what needs to be replaced. Caution: you will get wet doing this as I had to do the same once before.
Just a thought before you spring for a new pump.
