It very much depends on where precisely you live in the so-called West. It's not quite the same to be living in London or, conversely, in some provincial backwater somewhere, say in Nebraska (no offence intended). Don't tar all the inhabitants of the 'West' with the same brush (of ignorance).Like 99% or rather 99.9% of people on the West know only three Chinese cities at most: Beijing, Shanghai and Hongkong. Tianjin, Wuhan, Shenzen, Guangzhou, Nanjing, Chengdu, Chongqing etc are just some unknown Chinese words for them that could occur in some China-related news.
The chances are that if you ask your average Circle/District line passenger to name more than three Chinese cities (s)he will probably mention Chengdu or Chongqing (or perhaps Harbin and Nanjing too) but then many of these knowledgeable people are heading to their City jobs.