numbersix,
The most obvious entrances of Bowden Hill are at NS978748, above the footpath. There are several, maybe more than 10, along most of the north face of the hill. You will find them easily. I did look, and could not see any on the south side. I do believe that the limestone is continuous through the hill, but the south side is very steep, and I guess they had sufficient ventilation without going right through.
The west end looked older than the east, as there was much more weathering, and encroachment by undergrowth.
I have never been in these workings, as I was wearing my best clothes on the day of my visit, around 1969, but a quick look in the entrances suggested fairly extensive room and pillar workings. I must go back some time.
There are other limestone mines in the area, but I can't remember exactly where, but not far from NT004748. Under the road, if I remember correctly. And of course Hilderston is not too far.
Alan