IIRC the US army had horse cavalry ( what a name! ) in the Philipinnes and that will have been used to maintain colonial rule. I don't think they will have been of significant utility when war came and carrying horses to other fronts looks like waste. I imagine that mules might have been used for transport, perhaps in SE Asia and not in Normandy, but that's not cavalry. The British army used mules in Hong Kong until about 1970, iirc.
The Germans had horses in Normandy. A neighbor of my parents was a German who was called up during the war and survived amongst the horses, I think near Falaise, as one of the few men of his unit. But I think those horses were for transport too.