Horse-drawn armies need enormous amounts of supplies, the energy density of animal feed is a lot lower than that of oil. The British Army learned this lesson in WWI and was totally mechanized by 1939.
Depends where you are located and how large formations of troops you have concentrated on single location. Finnish Defense forces pretty much completely relied on horses during WWII and with great success. There are lot's of tales how Finns were able to supply very hard to get places and also had successful practice to bring back all the dead to be buried in their home towns.
They had to give up horses in the 60's because you could no longer find young men who could drive horse carts. The good mobility of horse in boggy and snowy terrain was only matched in the 80's with Bandvagn 206 carrier.
Yeah, but you can grow animal feed in the fields of occupied France, or take it from Ukranian peasants. You can't really grow petrol on those fields. (Although you can produce synthoil out of coal - which the Germans did.) The UK, the USSR, and the USA were, on the other hand, controlled something like 90% of all of the world's known oil reserves.
Fuel shortages for the fascists in WWII were so bad, in the later years of the war, they used horses to pull fighter planes from their hangars, to their runways. They did not have enough fuel to taxi under their own power.
