thirdcrank:
Well possibly, but I think that's just a bit too conspiratorial.
AIUI, the law just says that you can't drive a carriage on the footway (i.e. adjacent to a road). That law was not directed specifically at bikes or cars, as Parliament was mainly thinking of horse-drawn carriages and vehicles. IANAL, but I suspect that there are no recorded prosecutions of cyclists for pushing bikes on the pavement.
Bikes were invented before cars. There was no doubt in peoples' minds that this new and (to some) unnervingly rapid form of transport was a vehicle/carriage (which it is). Everyone therefore took it for granted that you couldn't take a carriage onto the footway. Cyclists had no great reason to wheel bikes on what were probably narrow pavements, and in many places pavements probably didn't exist anyway. Therefore all responsible cyclists wheeled their machines in the gutter while walking on the pavement (footway), and policemen shouted at the odd offender who blocked a narrow pavement by doing otherwise, but no-one got fined. Cyclists also parked in the gutter with one pedal placed on the pavement to hold up the machine.
This shared assumption changed much more slowly than did the number of cars, so by the 1960s we were still wheeling bikes in the gutter, even with constant lines of cars passing at 50mph (I exaggerate). At that point, however, the perceived safety issue and (possibly) the inconvenience to motorists of having to move out to overtake a "rider-less" machine became predominent in peoples' minds. They increasingly told their kids to wheel their bikes on the pavement. Also by this point you had numbers of parents who had never ridden bikes for transport, so the chain that passed down the instruction to "Wheel it in the gutter" had been broken.
Again, IANAL, but it is in that light that I read the summary of the judgement on
Crank v Brooks. Waller LJ is perfectly right that someone walking a bike must be a foot passenger. What that paragraph fails to address, however, is that the foot passenger is being accompanied by something that is obviously a vehicle!* That said, I am prepared to accept that the link gives a fair assessment of current legal opinion and practice (you won't get prosecuted for pushing a bike over a crossing) - it's just the reasoning that I find difficult.
I know that some people find this thread indescribably tedious, although confusingly it appears that they are still reading it

It interests me, however, because it is a clear example of shared assumptions being more important than what the law actually says. That's important when, instead of asking whether the law on vehicles applies to bikes on pavements, you ask whether the Highway Code advice and law on overtaking other vehicles applies to overtaking bikes. I perceive a gap there too between assumption and law
*This is rather like arguing that, having parked your car somewhere illegal, you are allowed to have it there because you are now standing beside it, and are therefore a pedestrian.