For someone to claim that the Polish government of 1939 wasn't fascist is extremely naive. While certainly not in the same league as Germany or even Italy, the Polish government was extremely nationalistic and thouroughly repressed all their minorities (except, ironically, the Germans, mainly due to fear I suspect). Despite this, the Poles were certainly the victims of 'realpolitik' and the Western Allies callously sold them out at every turn - and the Russians even worse of course. Despite this, the Poles put up quite a fight considering their resources. Note: the Germans also used cavalry in Poland (a brigade), as well as in every other campaign during the war (except North Africa, thought the Italians did use camel-mounted scouts), as did the Soviets and Axis allies in Russia.
Both the US and Great Britain had cavalry units during the war, though neither used them operationally (though the Brits did see some action in Palestine against Arabs and Jews).