I checked out the site, and found a good way to do this. KML is not the best format for handling attributes (as you found), but they also allow you to get the data as a CSV, which is a great format.
Download the CSV, in QGIS, choose Layer > Add Delimited Text Layer... Pick WKT as the geometry interpreter, and choose the field that says it has WKB geometry in it (which actually looks more like WKT anyway). I had not used this dialog before, and it's actually really great--more straight-forward and flexible than the one in ArcGIS.
That should do it. Once the layer is on the map you can save it as a shapefile if you like, and also assign the correct CRS to it.
EDIT: I found that the data from the site you linked is actually in WGS48 Web Mercator, EPSG 3857, not the national grid system as it claims.