Does the monarch ability transfer immediately when a player deals combat damage to the current monarch?

Yes — the transfer of the monarch designation is immediate upon combat damage being dealt. According to CR 718.3, whenever a player deals combat damage to the monarch, that player becomes the monarch. This triggered ability uses the word 'whenever,' meaning it triggers at the time combat damage is dealt and resolves as soon as players receive priority after that damage is dealt.

Specifically, combat damage is dealt in the combat damage step (CR 510.2). After damage is dealt, triggered abilities that triggered during that step are put on the stack and resolve before either player can take other game actions. The monarchy transfer trigger goes on the stack and, upon resolution, the attacking player becomes the new monarch (CR 718.3).

It does not matter how much damage is dealt — even 1 point of combat damage to the current monarch causes the transfer. The monarch status moves to the dealing player immediately upon that trigger resolving, not at end of turn or at any later point.

Concrete example: Alice is the monarch with 20 life. Bob attacks her with a 1/1 creature and Alice chooses not to block. In the combat damage step, that 1/1 deals 1 combat damage to Alice. The 'becomes the monarch' trigger fires, goes on the stack, resolves, and Bob is now the monarch — even though Alice still has 19 life remaining.

Note that this only applies to combat damage dealt to the monarch player. Noncombat damage (e.g., from spells or activated abilities) does not trigger the monarchy transfer (CR 718.3).