the cancelled person now says they can’t come and we’ve messed them around
That’s two separate reasons.
You’ve cancelled a booking, they’ve filled the slot with a different customer or otherwise made plans for their afternoon off. Seems perfectly reasonable, that’s what deposits etc are for.
They’re saying you’ve messed them around, well… cancelling but paying them anyway, that’s hardly messing them about, they’re quids in. Why would they say that? Did you get the arse with them when they said they couldn’t change their booking? Or are they just going “free money, lols, I’m off to the pub”?
Either way, it sounds like you’re looking for a new childminder. Either they don’t want you or you don’t want them.