I (British/Australian) use it, but not in a cultural sense. I use "the West" when talking about military or economic matters.

I generally prefer the term "Anglosphere" to refer to only the bits of "the West" that share that cultural viewpoint when I'm discussing cultural matters. It's not perfect, but it's useful.

Given the widening gap between the USA and Europe (and Canada) in economic and military matters, I'm not sure how much longer "the West" is going to be useful.