I've had both - the bottle thing I have from childhood and I did have the full hankie set of MGR hankies but no longer do. The latter I'm fairly sure probably went through the wash when I got them with no issues. I am unsure if I ever washed the hot water bottle case, but I used it a lot as a kid so would be surprised if Mum didn't.
Maybe a gentle wash..? Or handwash, as people suggest?