It was https://www.openstreetmap.org/way/501435865 that I believe is not a public bridleway. Looking at the PRoW overlay data and http://www.rowmaps.com/showmap.php?place=LU6%202GY&map=OSM&lat=51.865255&lon=0.537482&lonew=W the is no bridleway in this part of Dunstable Downs and so the cycleway should not be tagged as such with designated access tags.