UK Zebra Crossings confer no rights until you step off the pavement according to the Highway Code. I know in practice most drivers will identify people wanting to cross, but that's not what the book says.
It is however down to "sharing the roads" and considering the needs for others at the point of priority. Main roads have priority, but in general UK drivers will stop to allow a pedestrian to cross at the zebra crossing.
In Europe I have noticed a significantly higher number of zebra crossings in built up areas, but drivers ignore them and pedestrians cannot cross. Thereby making the point of lights and road markings irrelevant as they do not change driver behaviour.
This goes back to my point re it not being a matter of what a paper document states, but it should be a case of us all as road users just being excellent to one another.
In the past as a driver I have stopped on a main road to allow people to cross. As a pedestrian I have danced between moving traffic. As a cyclist I have passed queues, stopped for others, and been knocked off. As a driver I've slowed and flashed to allow somebody to turn when I could have carried on with priority, and at other times I have ploughed on with priority as had I slowed and flashed it could have caused confusion and a potential accident due to two vehicles moving off (and I know that by flashing I am simply "alerting another vehicle to my presence" or whatever the wording is, even though the general use in the world is "after you because I'm polite")