The Japanese made extensive use of bicycle troops in Malaya early in the war in the far east. On occasion, when ambushed and caught mounted, they got badly shot up, because the one thing you cannot do on a bicycle is use a weapon...
PS
Mounted cavalry did actually see a fair bit of action in WWII (not just Poland and the largely mythical lancer attacks on German Panzers) but especially on the Eastern Front where Germany, Italy and Russia all used cavalry (some of it armed with swords as well as rifles) and also in the Sino-Japanese war.
I also know of at least one cavalry charge by British led local mounted forces in Burma against the Japanese.