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I'm pretty sure cyclists are allowed to use zebra crossings without dismounting if both sides have a cycle path/ shared use path on
Cyclists are allowed to use zebra crossings but the point is whether (AIUI) cars are required to stop for cyclists who are riding (or even sitting upon) their bikes and want to use the crossing. I don't believe they are.
Car drivers would not be aware of whether there is a cycle/shared-use path either side of the crossing so I think that is a red herring.
The Crank vs Brooks court case shows that someone wishing to use the crossing and just happens to be pushing a bike should be treated as a pedestrian. (But such a ruling does not tell us anything about people who are riding their bikes at the time.)
The only certainty is that riding along and attempting to swing onto a zebra crossing and expecting all motor vehicles to stop and give way is a very risky strategy.
Oh, so there's some marginal excuse for the wankery I used to encounter on that stretch of road when I worked in Shepherds Bush? I'm still going to hate the tossers who would appear out of the blue and think the traffic would just stop while they zipped across the road.