Don't the British call it a "face flannel"? At least they do in the books. It was always "-rag" at our house. I had forgotten about the "rag bag". My mother always had one hanging in the pantry and I don't think a scrap of worn-out clothing, sheets, towels every got thrown away until it had recycled through the rag-bag.
In the bath, it's a washcloth, but in the kitchen, it's a dishcloth. They could be the same thing, but the ones we use in the bath are thicker and larger than the ones we use for dishes. Dishrag wouldn't be at all foreign to me, however. Growing up, I think they were washcloths and dishrags. I have rags, but I use them for things like cleaning spilled gasoline or oil off of the lawnmower.
I call them a bath cloth and a dish cloth. Face flannel? ????? My, those Englishters have some strange words.
I grew up with rooms I was not even supposed to go in! Also could not use the towels and rags for company also a few dishes and cups.
I think just about everyone had a "fancy" set of towels back then hanging in the main bathroom that nobody.on.the.pains.of.death could touch. I read a funny anecdote about a woman who was getting ready for a party the next day. She hung the fancy towels in the hall bathroom and pinned a note on them for her family's benefit: "TOUCH THESE AND YOU'RE DEAD!!!" The next night after the party, she was tidying up and went into the bathroom. Oops, she had forgotten to remove the note. #cringe#