Strong winds from caves are usually generated in three ways: wind blowing into another entrance, the cave changing the air density and sucking air through, or meteorological air pressure changes filling and emptying the cave*1. As will become apparent, the Llangattock Escarpment is a perfect place to look for caves with a strong breeze caused by at least a couple of these mechanisms. First let's look at the least likely mechanism.
The simplest mechanism for a strong wind to develop is to have a cave which has more than one entrance and for wind to blow straight through it. However, the description explicitly states that the wind occurs even when there aren't any other winds, so this is unlikely here. In the case of larger caves, the friction in narrow parts of the system and air storage in larger parts makes the progress of such a wind through the system more complex, and reduces its strength. Instead, larger winds in more complex systems tend to be the result of pressure differences between the cave and the external environment. These can either be due to local temperature changes or broader meteorological systems.
More on complex cave winds