Annihilator abilities trigger and resolve during the declare attackers step. The defending player chooses and sacrifices the required number of permanents before they declare blockers. Any creatures sacrificed this way won't be able to block.

If a creature with annihilator is attacking a planeswalker, and the defending player chooses to sacrifice that planeswalker, the attacking creature continues to attack. It may be blocked. If it isn't blocked, it simply won't deal combat damage to anything.

The last ability granted by Nazgûl Battle-Mace triggers whenever an opponent sacrifices a nontoken permanent for any reason, not just due to the annihilator ability.

If an opponent sacrifices a nontoken permanent, it doesn't matter which graveyard it goes to. The last ability granted by Nazgûl Battle-Mace will trigger.

If the card leaves the graveyard before the last ability granted by Nazgûl Battle-Mace resolves, the opponent will still have the option to pay 3 life. Regardless of their choice, you can't put the card onto the battlefield, even if it returns to the graveyard before the ability resolves.

If a replacement effect, such as that of Rest in Peace, causes a sacrificed nontoken permanent to be put into a different public zone than a graveyard, the last ability granted by Nazgûl Battle-Mace will still trigger and you will still put the card onto the battlefield under your control if its original controller doesn't pay 3 life.

If an opponent sacrifices a nontoken permanent as part of paying the cost of a spell or ability, the last ability granted by Nazgûl Battle-Mace will trigger and go on the stack on top of that spell or ability. The ability granted by Nazgûl Battle-Mace will resolve before that spell or ability.

If you put an Aura on the battlefield using the last ability granted by Nazgûl Battle-Mace, you choose what the Aura will enchant just before it enters the battlefield. An Aura put onto the battlefield this way doesn't target the player or permanent it will enchant, so permanents or players with hexproof may be chosen; however, the chosen recipient must be able to legally be enchanted by the Aura, so a player or permanent with protection from one of the Aura's qualities can't be chosen this way. If there's nothing legal for the Aura to enchant, it stays in the graveyard.