Deary, deary me. This cycling 'off-road' is muddy issue indeed. I have Googled and read two informed sources, but I shall have to leave you all to make your own mind up as to how useful an exercise it is.
http://www.bikehub.co.uk/featured-articles/cycling-and-the-law/
http://www.guardian.co.uk/environment/bike-blog/2011/jul/28/cycling-pavement-offence
It seems that as far as pedestrianised areas are concerned Local Government:by-laws and signage,is where the action should be. In any event children under 16 will not be prosecuted.
This is yet another symptom of what has to be Britain's Disease, 'Respectlessness'.