I think most Americans who read and pay attention to media are pretty familiar with differences in British terminology for certain common words, including "lift," "lorry" etc. As for saying "lift" to mean "ride," think context would make it pretty clear. Heck, we have different colloquialisms in parts of the US too.
When a particular British expression is unfamiliar, most Americans certainly know they use different words for some things and are capable of grasping meaning via context. I learned the word "spanner" = "wrench" while playing a video game, actually. And when Harry Potter laced up his "trainers," I figured out they were some kind of athletic shoe.
I'd actually find it really weird to read a novel set in the UK where terminology was "translated" to US usage, or to have a British character in a novel who used American words for things.