It is probably not common in places like London where the roads can be "American-wide", but definitely common in other parts of UK.
Uh, noo no nono no.
London? American-wide streets? Its the exact opposite. London is so ancient and has been left so unaltered since basically after the Great Fire in the 1600s (!) that no, the streets are mostly horrendously narrow.
There are parked cars on most roadways and yes, some of them have to be two wheels on pavements in some spots. You see that a lot around central and even not-so central London. It also depends on the particular district/borough. I'm in a spot in North London riddled with the most narrow lanes -- there is even a famous spot that is so protected for being historical that the two-way traffic throughway is
one car wide and everyone has to employ a MASSIVE amount of etiquette just to all get through!
But drive out just a matter of three miles more northward, and you get to dual carriageways and wider streets some of which have more of an American width, yes. Outer London is hit and miss -- some wide roads, some still as narrow as they were in 1600 or "pick-a-year-it-ain't-changed"...
However, even with all the narrow stuff going on, no, it's pretty rare to see pedestrians having to dodge for their lives while cars are mounting pavements all over the place. It's misleading to say it's as bad as all that.
At the same time, as a pedestrian in a built up and heavily congested place like most of London is, you SHOULD be wary even if you have the right of way -- don't just dash out in the street, because if you have to cause a driver to slam on the brakes, there is usually going to be a stream of tailgating vehicles behind that one car and you will cause a wreck. Both drivers AND pedestrians really need to consider EACH OTHER because it goes both ways, the careless behavior.