It would be best to clarify this statement.
This is true of most footpaths, but where the land owner e.g. the Malvern Conservators has managed to get a specific bylaw in place, then it IS illegal to ride on the footpaths.
Technically, that would be a “MAYBE” rather than an “IS” – it is only criminal because the landowner has NOT giver permission, however it remains within the landowners powers to grant authority. A perfect example would be the Forestry Commission, where riding on any FC land is technically illegal under the byelaws, but the FC have given extensive permission to use the land regardless of this.
In the case of common land, the landowner has the authority to give permission to do anything they want on the land, as long as it does not interfere unacceptably with the rights of the commoners, or is restricted by law (eg, erection of fenced enclosures on common land) – given that a great many commons are already subject to S193 of the law of property act (unrestricted access for air and exercise on foot or horseback) or foot access under CROW, the argument that giving permission for bikes would interfere with the rights of commoners (if any exist) is at best spurious.