I've heard "bum a ride" here in America, although it's a rare word in a rare phrase.
I'll disagree about it being rare; I hear the phrase at least a few times per week (I work in a place where people frequently ask other people for rides, so perhaps that's relevant)
The other day I realized I hadn't eaten basic tier fast food in 15 years. That could be anecdotal evidence for some general trend ("fast casual! increasing consumer health consciousness!"), or it could just be me.
I'd be willing to bet a totally irresponsible amount of money that if I went out to bars in Atlanta tonight, I would hear someone say "bum a cigarette" before closing time rolled around.