The timestamp of Fealty to the Realm's control-changing effect is the time at which it entered the battlefield and won't change when another player becomes the monarch. This means that, if another player gains control of the creature enchanted with Fealty to the Realm, they won't lose control of it when another player becomes the monarch. When (or if) that control-changing effect ends, the enchanted creature will then be controlled by the current monarch, which might be a different player than it was when that control-changing effect began.