HWC 79 of the Highway Code states that cyclists should dismount and walk across zebra crossings.
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Rule 79 of the current highway code is about roundabouts. If you are a driver, keeping up to date with changes in the highway code is a requirement of the national standards and your driving licence. You can find the real UK highway code at
https://gov.uk/highway-code - don't be taken in by internet fakes trying to either cause conflict or sell driving aids.
Rule 81 includes "Do not ride across a pelican, puffin or zebra crossing. Dismount and wheel your cycle across" but this is not a MUST (so there's no law requiring it) and the code authors haven't thought through the consequences of that instruction. Pedestrians will not appreciate having cyclists high-kicking to dismount and mount in the usually-limited space at either side of a crossing, or them taking up more space on the crossing for longer. Also, many puffin crossings have cycle symbols on their green signals, so the above instruction is simply wrong in those cases. That rule needs rewriting, but nobody at DfT cares enough about cycling to do so.
Although not a legal offense in itself, failing to dismount could be used as evidence of dangerous or careless cycling if an accident occurs.
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It could, although it's unlikely, but it's really better not to collide with anyone if you can avoid it. The act of trying to avoid a collision is likely to suggest you're not being dangerous or careless.
Plus, motorists must give way to a pedestrian on a zebra crossing , but are not required to give way to a cyclist.....
In general, no. This is almost certainly the real reason why the instruction not to ride across remains in the rule. It's the same sort of bottom-covering that resulted in "CYCLISTS DISMOUNT" at every side road and driveway until the highway code was clarified: drivers used to be told explicitly to give way to pedestrians crossing side roads, but not cyclists.
However, the current highway code does actually say drivers MUST give way to cyclists on zebra as well as parallel crossings (rule 206) but I think that may be an editing error or misleading simplification. And if the zebra is across the mouth of a side road (as seen in parts of London and elsewhere), any turning driver should give way to cyclists as if the zebra wasn't there (rule H3).