IANAL, but AIUI UK case law (Crank v Brooks) is that a person pushing a bike on foot is a pedestrian (well, technically a 'foot-passenger', but they're not considered to be driving or riding it.)
But that ruling refers to the status of the person not the presence or absence of a bicycle.
The original decision was that a person pushing a bicycle across a pedestrian crossing was not a foot-passenger and therefore did not need to be accorded precedence at the crossing, i.e. the presence of the bike changed their status.
The appeal (which is
Crank vs Brooks) reversed this and said that a person pushing a bicycle across a pedestrian crossing should be considered a foot-passenger, but they're still a foot-passenger pushing a bicycle.
The point is the
Crank vs Brooks ruling doesn't make the bicycle disappear, it reinforces the status of the person irrespective of whether a bicycle is present.
If bikes (even pushed) are restricted somehow (e.g. a TRO) then that's what is going to apply. The question in the restriction is not the status of the person but the presence of a bicycle.
Crank vs Brooks doesn't change anything or help in this case.
You can't try and take a bicycle into a cinema with you and just claim
Crank vs Brooks. (I know it's not quite the same thing as a cinema isn't a public place, but you get the idea.)
