Wave model of light and time delayWatch
The intensity of a wave is the power (energy per second) transferred per unit area. If the intensity is greater, it takes less time for the same amount of energy to be transferred to the same area.
You asked what happens if you increase the intensity of the light.
The answer is, if you do that, you provide more energy per second and therefore would reduce the expected delay time. Of course this doesn't happen, hence the photon theory of light.
What you need to realise, is that according to the wave theory, the energy transferred per second depends on the intensity. Indeed, that is the definition of intensity. So if you increase it, irrespective of whatever frequency the light is, you provide more energy and would expect either a shorter delay (there was never a delay) and the emitted electrons to have more energy. (They didn't, there were more of them, though.)