"Did the S.A.S Desert Raiders have Training shoes during the last War?................"
Well, they, and their colleagues in the Long Range Desert Group, Popski's Private Army etc, often wore, appropriately enough, DESERT BOOTS. These were lightweight, sort-of-crepe soled, ankle-height numbers.
Sometimes also called "brothel-creepers". (a bit quieter than hob-nailed boots on the marble floors, apparently)