I know that the US army was fully mechanised and didn't use any horses but what about the British
I'll let someone more qualified discuss the level of mechanization in the British army, but I did want to point out that your assumption that the US military was fully mechanized in the second world war is an erroneous one:
The US military made use of horses for supply, particularly in the Pacific, and the 26th Calvary Regiment actually had the last calvalry charge in US Army history in combat at Bataan in 1942.
The history of the US Veterinary Corps notes 56 000 horses and mules under their charge at peak numbers during world war two. ^^1
https://history.amedd.army.mil/booksdocs/wwii/vetservicewwii/chapter14.htm