Kipling is saying that if 'an English Cavalry regiment cannot run' (run away in fear from something such as an enemy) is literally true, then the soldiers of the White Hussars ceased to be English Cavalry soldiers for two hours (and thus were humorously supposed by Kipling to have vanished from the Army List during that time).
A bridle is part of the equipment used on a horse to enable the rider to control it.
The phrase 'that ever drew bridle' is a humorous play on the set expression 'that ever drew breath'. To draw breath is to be alive or exist; 'the best person who ever drew breath' means 'the best person who ever lived'. 'The best Regiment that ever drew bridle' means 'the best Regiment that ever rode on horseback'.
The Army List is an official publication which lists the serving officers of the British Army. It has existed in various forms since 1702.
The officers of the White Hussars ceased to be deserving of that title, Kipling suggests sarcastically, (they allowed their men to panic) for (during a period of) two hours. Kipling imagines that, because English Cavalry Regiments supposedly 'cannot run', when the men ran, they ceased to be soldiers of the White Hussars temporarily, and the officers, listed in the Army List, must have figuratively vanished from it for those two hours. You will discover the circumstances as you read the story.