It's hard for me to imagine guns or imperial units catching on any time soon (to be fair, we didn't invent imperial units!), but I would not be too surprised if some European nations start slowly privatizing bits of their healthcare systems. The profit motive is sneaky and creeps in slowly. Usually it starts by capturing the hearts and minds of those in power.

And regarding attitudes toward women's bodies, one only needs to visit any beach in southern Spain and count up the topless women to notice those values are already Americanizing.