The situation in the New Forest is unusual. Many of the ways tagged highway=bridleway have been tagged this way because they can be used by equestrians. However, they are not public bridleways and you are not permitted to cycle on them unfortunately (thank the New Forest Verderers...). There are very few public footpaths and public bridleways in the New Forest because you can walk and ride a horse most places. Unfortunately this makes it worse for cycling off road than many places. There are some designated cycle tracks, which are almost all gravel. There are a few bridleways, restricted byways and byways here and there. Personally I'm not that keen on tagging these paths as bridleways but I'm not going to go round changing them all as it's not unreasonable.