As stated in the notes at the bottom of the first linked page, (copied below) these are the rules for wildfowling with a shotgun below the High water mark, and as such have nothing to do with air guns
the angling rules might be more relevant as air gun pellets would all be inside the prohibited weight range.

I still can't find the thing I read, but I only stumbled across it by accident, until I read it I had no idea there was any issue.

EXPLANATORY NOTE
(This note is not part of the Regulations)

These Regulations (“the Regulations”), which apply to England only, prohibit the use of lead shot for shooting with a shot gun–

(a)on or over any area below the high-water mark;
(b)on or over the sites of special scientific interest included in Schedule 1 to the Regulations;
(c)any wild bird included in Schedule 2 to the Regulations. These birds are ducks, geese and swans (all species of each); coot; moorhen; golden plover and common snipe (regulation 3).
“Lead shot” means any shot made of lead or of any alloy or compound of lead where lead comprises more than 1% of that alloy or compound (regulation 2).