I call this the Old Oil Depot. It's situated at the edge of the common, right next to a road.
In my Google Earth image of the common on the Greenham home page you can just make out the concrete circle at the extreme right
of the image, just below the oval running track that is visible, which shows the overall location of the object.
I don't think there's any doubt that it was the point where deliveries of jet and other fuels were taken delivery of. The fixtures and fittings are in excellent condition, considering their age and the ravages of time and rain, not to mention metal thieves, who have stolen such odd bits and pieces that they could detach. Apart from the object in the first two pictures after the overhead image, all the objects you see here are situated on top of a large concrete circle, and I have no doubt that underneath this circle must lie an enormous storage tank. Who knows?