The co-ordinate systems are the same but you have to bear in mind that anything small depicted on the map will not be shown precisely (footpaths are a good example of this - the OS try to ensure that footpaths show on the correct side of boundaries but a footpath was typically less than 2ft wide and a wall might be less scale those down to 1k25 - both marked and good enough for most general purposes).
Back to the gated entrance:
View attachment 1289
The numbers, bottom left are the 1m precision location of the entrance (for those who prefer the more common LLnnnnnn type format SH 31861 26504) not the map centre.
Note the three shafts (again, ignoring the size they are shown at as they can't be that big).
The latest 1m LIDAR probably wouldn't help much but typically there is a chunk missing which is just the bit we'd want. The 50cm LIDAR is only available for a few areas so far and obviously not this bit that we need! If available it might show other potential things that are hidden by foliage on the various aerial imagery available.
On to the 'trial adit'. Again, numbers bottom left are the adit not map centre. In this case the 'Issues' is significant as it means there is an opening of some sort from which water is coming.
View attachment 1290
So that leaves the 'Old Day Adit'. Looking at the first screengrab the location from the co-ordinates posted by Dave would put it left and above the leftmost shaft, around the corner of that same plot. Nothing shown on Mastermap but that isn't surprising as there would have to be recent arial data and the scar seen. Of course, because of the inherent imprecise nature of a marking of 'level' on an old map (remember, it is small so can't be marked precisely due to the map scales used in over the counter mapping) we'd need to look around a slightly bigger area on the ground.