Supposedly, I am a coordinate systems expert, and it does look like there's a missing transformation.
Both posted shapefiles are still using British National Grid (based upon OSGB 1936).
If I add "Imagery" basemap which is using WGS 1984 Web Mercator, then add the shapefiles, I get the warning message about different geographic coordinate systems. I then set a transformation from OSGB 1936 to WGS 1984 using:
OSGB_1936_To_WGS_1984_Petroleum
I don't know if this transformation is available in the version you're using.
When I okay that dialog and compare the Shannabank Hill, Fort poly/point against the imagery, I can see that they're overlaying within a few meters. If I use OSGB_1936_To_WGS_1984_1 (a not-as-accurate transformation), the offset is more, closer to 7-8 meters as opposed to 4m.
When you converted the shapefiles to WGS84, did you use the Project Tool or the Define Projection tool or the data's property page in ArcCatalog? The latter two update the coordinate system information only, they don't change the coordinate values to the 'new' coordinate system.
Melita