If combat damage dealt to the monarch causes that player to lose the game, the triggered ability that causes the controller of the attacking creature to become the monarch doesn't resolve. In most cases, the controller of the attacking creature will still become the monarch as it is likely their turn.
If the triggered ability that causes the monarch to draw a card goes on the stack and a different player becomes the monarch before that ability resolves, the first player will still draw the card.
If the monarch leaves the game during another player's turn, that player becomes the monarch. If the monarch leaves the game during their turn, the next player in turn order becomes the monarch.
Magic: The Gathering · ™ & © Wizards of the Coast · Illus. Lorenzo Mastroianni
Card data via Scryfall. Rarix is fan-made; not endorsed by the rights holders.
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.
If the monarch leaves the game, and that player is the owner of Fealty to the Realm, the active player or the next player in turn order becomes the monarch. That player gains control of the enchanted creature for a very brief moment. At the same time, Fealty to the Realm leaves the game, and control of the enchanted creature reverts to its owner (unless another control-changing effect applies.)
If the monarch leaves the game, and that player is not the owner of Fealty to the Realm, the active player or the next player in turn order becomes the monarch and thus gains control of the enchanted creature.
The game starts with no monarch. Once an effect makes one player the monarch, the game will have exactly one monarch from that point forward. As a player becomes the monarch, the current monarch (if any) ceases being the monarch.
There are two inherent triggered abilities associated with being the monarch. These triggered abilities have no source and are controlled by the player who was the monarch at the time the abilities triggered. The full texts of these abilities are "At the beginning of the monarch's end step, that player draws a card" and "Whenever a creature deals combat damage to the monarch, its controller becomes the monarch."
In the case where Fealty to the Realm is enchanting a creature but there is no monarch, the second ability will create a control-changing effect with the timestamp mentioned above, but that effect won't do anything until a player becomes the monarch.
