Ivykaty - from your link
"Paths away from the road
So if your path is away from the road, does this mean you can definitely cycle on it?
Unfortunately not ? there might be other specific rules banning it. For example, cycling on your path might be banned by local byelaws, and breaching them may be an offence ? this is especially likely in parks and pedestrianised zones, for example.
Alternatively if the path is on private land, and you use it in a way (like cycling) which hasn?t been permitted by the landowner or a public right of way, you might be trespassing ? which is a civil wrong for which you can be sued (although it?s not an offence, and you can?t be prosecuted).
So it?s very difficult to know in advance whether you?re allowed to cycle on a path which is away from the road. As a general rule there should normally be signs telling you what?s allowed, and your best bet will just be to obey them.
As for specific kinds of country paths ? bridleways, public footpaths and the like ? that?s a whole other post."
I'm not sure where this says that cycling on a public footpath is permissible? The link that I made (to an environmental law site, not a walkers site or a cyclists site) specifically indicates that cycling on a public footpath is not allowed. Your link suggests that for country paths, such as public footpaths - 'that's a whole other post' - not that its legal.