DEFRA's scope includes littering, dogs at large, etc. AFAIK, it does not include cycling on pavements - that is the DfT's turf, and their guidance from the Home Office is, "...the issue is about inconsiderate cycling on the pavements...the fixed penalty needs to be used with a considerable degree of discretion and it cannot be issued to anyone under the age of 16."
You also said, "...there is in general no minimum age at which an FPN can be issued...", but I think it is well-established in law that no child under 10 can be legally sanctioned at all.